Divide using synthetic division. (x⁴+3x³+3x²+4 x+3) / (x+1) .

Step-by-step explanation:

c = cost of the camera

 6.5 % of 'c' is  $78

.065 * c = $ 78

c = $78 / .065 = $ 1200

(a) The relation represented by the zero-one matrix is not a partial order because it is not reflexive.

(b) The relation represented by the zero-one matrix is a partial order because it is reflexive, antisymmetric, and transitive.

(c) The relation represented by the zero-one matrix is not a partial order because it is not antisymmetric.

(a) For a relation to be a partial order, it needs to satisfy three properties: reflexivity, antisymmetry, and transitivity. Reflexivity means that every element is related to itself. In the given zero-one matrix, there is a zero on the main diagonal, which indicates that not every element is related to itself. Therefore, the relation is not reflexive and, as a result, cannot be a partial order.

(b) In the second zero-one matrix, every element is related to itself as indicated by the ones on the main diagonal. This satisfies the reflexivity property. Antisymmetry means that if two elements are related in one direction, they cannot be related in the opposite direction, except when they are the same element.

The matrix satisfies this property as there are no pairs of elements that are related in both directions, except for the self-relations. Lastly, the matrix satisfies the transitivity property, which means that if element A is related to element B and element B is related to element C, then element A is also related to element C. Since all three properties are satisfied, the relation represented by the zero-one matrix is a partial order.

(c) In the third zero-one matrix, there are pairs of elements that are related in both directions, which violates the antisymmetry property. This means that the relation is not antisymmetric and, consequently, cannot be a partial order.

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Answer:

C) 3

Step-by-step explanation:

To find the slope given a table with points, use the formula:
[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

Use the points:

(-2,-2) and (-1,1)

[tex]\frac{1+2}{-1+2}[/tex]

simplify

3/1

=3

So, the slope is 3.

Hope this helps! :)

An example of a pair of angles that satisfies the given condition of "two obtuse adjacent angles" is Angle A and Angle B, where Angle A and Angle B are adjacent angles and both are obtuse.

Adjacent angles are two angles that share a common vertex and a common side but have no common interior points.

Obtuse angles are angles that measure greater than 90 degrees but less than 180 degrees.

To meet the given condition, we can consider Angle A and Angle B, where both angles are adjacent and both are obtuse.

Since the condition does not specify any specific measurements or orientations, we can assume any two adjacent obtuse angles to satisfy the condition.

For example, let Angle A be an obtuse angle measuring 110 degrees and Angle B be another obtuse angle measuring 120 degrees. These angles are adjacent as they share a common vertex and a common side, and both angles are obtuse since they measure more than 90 degrees.

Therefore, Angle A and Angle B form an example of a pair of "two obtuse adjacent angles" that satisfies the given condition.

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Mathematics is an interesting subject that is constantly evolving and changing. Researching different areas of Mathematics Teaching can help to advance teaching techniques and increase the knowledge base for both students and teachers.

There are several researchable areas of Mathematics Teaching. One area of research is in the development of new teaching strategies and methods.

Another area of research is in the creation of new mathematical tools and technologies.

A third area of research is in the evaluation of the effectiveness of existing teaching methods and tools.

A fourth area of research is in the identification of key skills and knowledge areas that are essential for success in mathematics.

Finally, a fifth area of research is in the exploration of different ways to engage students and motivate them to learn mathematics.

Overall, there are many different researchable areas of Mathematics Teaching.

By exploring these areas, teachers and researchers can help to advance the field and improve the quality of education for students.

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The best answer regarding the effects of carbon monoxide is option c, CO results in less oxygen loading hemoglobin but unloading is not changed.

Carbon monoxide binds up more tightly to the hemoglobin as compared to the oxygen molecules. This reduces the oxygen-carrying capacity of the blood and results in less oxygen loading onto hemoglobin.

However, once oxygen is already bound to hemoglobin, CO does not significantly affect its release or unloading. Therefore, option c is the most accurate statement among the given choices.

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a) The Redemption value of the issued bonds redeemable at par is P50,000.

b) The annual deposits required to meet the requirements of the sinking fund at the end of the 15th year is  b. P2,460.

c) The principal in the fund at the end of the 12th year is d. P37,520.

The annual deposits can be determined using an online finance calculator as follows:

The number of bonds issued = 50

The face value (par value) per bond = P1,000

Redemption period = 15 years

a) Redemption value of the bonds = P50,000 (P1,000 x 50)

N (# of periods) = 15 years

I/Y (Interest per year) = 4%

PV (Present Value) = P50,000

FV (Future Value) = P0

Results:

b) Annual Deposit = P2,460

Sum of all periodic payments = P36,900

Total Interest = $13,100

c) Amount at the end of 12th year = P37,520

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The redemption value of the bonds is P50,000. The annual deposits into the sinking fund are P2,566. The principal in the fund at the end of the 12th year is P43,270.

To find the redemption value, we multiply the number of bonds (50) by the face value of each bond (P1,000), giving us a total of P50,000.

To calculate the annual deposits into the sinking fund, we need to determine the amount needed to accumulate P50,000 at the end of 15 years with an interest rate of 4%. This can be done using the future value of an ordinary annuity formula.

The formula is:

A = P * [(1 + r)^n - 1] / r,

where A is the desired future value, P is the annual deposit, r is the interest rate, and n is the number of years.

Plugging in the values, we have:

P = 50,000 * (0.04) / [(1 + 0.04)^15 - 1] = P2,566.

Therefore, the annual deposits into the sinking fund are P2,566.

To find the principal in the fund at the end of the 12th year, we can use the future value of a single sum formula:

FV = PV * (1 + r)^n,

where FV is the future value, PV is the present value (initial principal), r is the interest rate, and n is the number of years.

The principal in the fund at the end of the 12th year is calculated as:

PV = 2,566 * [(1 + 0.04)^12] = P43,270.

Therefore, the principal in the fund at the end of the 12th year is P43,270.

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The solution to the problem is the simplified expression: 5x³ - x² - 3x + 13.

To solve the given problem, you need to simplify and combine like terms. Start by adding the coefficients of the same degree terms.

(3x³ - x² + 4) + (2x³ - 3x + 9)

Combine the like terms:

(3x³ + 2x³) + (-x²) + (-3x) + (4 + 9)

Simplify further:

5x³ - x² - 3x + 13

In this expression, the highest power of x is ³, and the corresponding coefficient is 5. The term -x² represents the square term, -3x represents the linear term, and 13 is the constant term. The simplified expression does not have any like terms left to combine, so this is the final solution.

Remember to check for any specific instructions or constraints given in the problem, such as factoring or finding the roots, to ensure you address all requirements.

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The minimum length of the bolt required to bolt the two pieces of wood together is 2 inches.

The minimum length of the bolt needed to bolt two pieces of wood together is 2 inches. Here's how to arrive at the answer:Given that one piece of wood is 1/2 inch thick and the second piece is 5/8 inch thick. The thickness of the washer is 9/16 inch, while the nut is 3/16 inch thick.

We need to find the minimum length (in inches) of bolt required to bolt the two pieces of wood together.Using the formula for the minimum length of bolt needed to bolt two pieces of wood together, we can express it as:

Bolt length = thickness of first piece + thickness of second piece + thickness of the washer + thickness of the nut+ extra thread required for a secure hold

The extra thread required for a secure hold is 3/4 inch, that is 1/2 inch for the nut, and 1/4 inch for the thread on the bolt.

Total thickness = 1/2 inch + 5/8 inch + 9/16 inch + 3/16 inch + 3/4 inch (extra thread)= 2 inches

Therefore, the minimum length of the bolt required to bolt the two pieces of wood together is 2 inches.

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The expression given can be expressed in it's splest term as 5 : 3

Given the expression :

To simplify to it's lowest term , divide both values by 44

Hence, we have :

At this point, none of the values can be divide further by a common factor.

Hence, the expression would be 5:3

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The statement is true.

The average length of time between successive events of a given size (or larger) is indeed referred to as the recurrence interval (RI).

To understand this concept better, let's break it down:

1. Recurrence Interval (RI): The recurrence interval is a measure used in statistics and probability to determine the average time between events of a specific size or larger.

It is commonly used in fields such as hydrology, seismology, and finance to analyze the frequency and magnitude of events.

2. Successive Events: In this context, successive events refer to events that occur one after the other, without any gaps in between.

For example, if we are studying earthquakes, successive events would be the occurrence of earthquakes of a certain magnitude within a specific area.

3. Given Size or Larger: The recurrence interval focuses on events of a given size or larger. This means that we are considering events that meet or exceed a particular threshold.

For instance, if we are analyzing rainfall patterns, we might be interested in the recurrence interval of rainfall events that exceed a certain amount, such as 1 inch or more.

To illustrate this concept, let's consider an example:

Suppose we are studying hurricanes in a coastal region. We want to determine the average length of time between Category 3 or higher hurricanes.

We collect data and find that, on average, there is a Category 3 or higher hurricane every 5 years.

In this case, the recurrence interval (RI) for Category 3 or higher hurricanes would be 5 years. This means that, on average, we can expect a Category 3 or higher hurricane to occur once every 5 years in that coastal region.

To summarize, the statement is true: the average length of time between successive events of a given size (or larger) is referred to as the recurrence interval (RI).

It helps us understand the frequency and timing of specific events in various fields of study.

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If the function is f(x, y, 3) = xy₂ x ² + 2²-5 хуе 4, the gradient of the point (1,3,-2) is (-204, -36, -324).

We need to calculate the gradient of the point (1,3,-2). The gradient is the rate of change of a function. It is also called the slope of a function. The gradient of a point on a function is defined as the derivative of the function at that point. In three dimensions, the gradient of a point is a vector with three components.

Each component of the gradient is the partial derivative of the function with respect to one of the variables. The gradient of f(x, y, z) at a point (x0, y0, z0) is grad f(x0, y0, z0) = ( ∂f/∂x, ∂f/∂y, ∂f/∂z )at the point (x0, y0, z0)

We have the function is f(x, y, 3) = xy₂ x ² + 2²-5 хуе 4

The partial derivatives of the function are as follows:

∂f/∂x = yz³ + 2x - 5y²z³∂f/∂y

= xz³ - 10xyz²∂f/∂z

= 3xy²z²

Using the above formula for calculating the gradient, we get

grad f(x, y, z) = ( yz³ + 2x - 5y²z³, xz³ - 10xyz², 3xy²z² )

The gradient of the point (1,3,-2) is :

grad f(1,3,-2) = ( 3×(-2)³ + 2×1 - 5×3²(-2)³, 1×(-2)³ - 10×1×3²(-2)², 3×1×3²×(-2)² )

= ( -204, -36, -324 )

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Answer:

The "Federal Triangle" is formed by the end points of the White House, the Washington Monument, and the Capitol Building. These points are also based on the Pythagorean Theorem of right angle triangles. Symbolically, the vertical line between the White House and the Washington Monument represents the Divine Father.

Answer:

26ocm

Step-by-step explanation:

you do 2 plus 4 plus 5.

Answer:

[tex] {c}^{ \frac{1}{2} } = \sqrt{c} [/tex]

[tex] {c}^{ \frac{2}{3} } = \sqrt[3]{ {c}^{2} } [/tex]

[tex] c = {2}^{6} = 64[/tex]

there is one tangent line to the curve y = x/(x+2) that passes through the point (1, 2), and it touches the curve at the point (-2, -1).

To find the number of tangent lines to the curve y = x/(x+2) that pass through the point (1, 2), we need to determine the points on the curve where the tangent lines touch.

First, let's find the derivative of the curve to find the slope of the tangent lines at any given point:

y = x/(x+2)

To find the derivative dy/dx, we can use the quotient rule:

[tex]dy/dx = [(1)(x+2) - (x)(1)] / (x+2)^2[/tex]

      [tex]= (x+2 - x) / (x+2)^2[/tex]

     [tex]= 2 / (x+2)^2[/tex]

Now, let's substitute the point (1, 2) into the equation:

[tex]2 / (1+2)^2 = 2 / 9[/tex]

The slope of the tangent line passing through (1, 2) is 2/9.

To find the points on the curve where these tangent lines touch, we need to find the x-values where the derivative is equal to 2/9:

[tex]2 / (x+2)^2 = 2 / 9[/tex]

Cross-multiplying, we have:

[tex]9 * 2 = 2 * (x+2)^2[/tex]

[tex]18 = 2(x^2 + 4x + 4)[/tex]

[tex]9x^2 + 36x + 36 = 18x^2 + 72x + 72[/tex]

[tex]0 = 9x^2 + 36x + 36 - 18x^2 - 72x - 72[/tex]

[tex]0 = -9x^2 - 36x - 36[/tex]

Simplifying further, we get:

[tex]0 = 9x^2 + 36x + 36[/tex]

Now, we can solve this quadratic equation to find the values of x:

Using the quadratic formula, x = (-b ± √([tex]b^2[/tex] - 4ac)) / (2a), where a = 9, b = 36, c = 36.

x = (-36 ± √([tex]36^2[/tex] - 4 * 9 * 36)) / (2 * 9)

x = (-36 ± √(1296 - 1296)) / 18

x = (-36 ± 0) / 18

Since the discriminant is zero, there is only one real solution for x:

x = -36 / 18

x = -2

So, there is only one point on the curve where the tangent line passes through (1, 2), and that point is (-2, -1).

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There are two tangent lines to the curve y=x/(x+2) that pass through the point (1,2) and they touch at points (0,0) and (-4,-2). This was determined by finding the derivative of the function to get the slope, and then using the point-slope form of a line to find the equation of the tangent lines. Solving the equation of these tangent lines for x when it is equalled to the original equation gives the points of tangency.

To find the number of tangent lines to the curve y=(x)/(x+2) that pass through the point (1,2), we first find the derivative of the function in order to get the slope of the tangent line. The derivative of the given function using quotient rule is:

y' = 2/(x+2)^2

Now, we find the tangent line that passes through (1,2). For this, we use the point-slope form of the line, which is: y- y1 = m(x - x1), where m is the slope and (x1, y1) is the point that the line goes through. Plug in m = 2, x1 = 1, and y1 = 2, we get:

y - 2 = 2(x - 1) => y = 2x.

Now, we solve the equation of this line for x when it is equalled to the original equation to get the points of tangency.

y = x/(x+2) = 2x => x = 0, x = -4

So, there are two tangent lines that pass through the point (1,2) and they touch the curve at points (0,0) and (-4, -2).

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Last year, Juan had $10,000 to invest. He decided to divide his investment into two accounts: one that paid 9% simple interest per year and another that paid 7% simple interest per year. After one year, Juan received a total of $740 in interest. Juan put $2,000 and $8,000 into the account that offered 9% and 7% interest, respectively.

To find out how much Juan invested in each account, we can set up a system of equations. Let's say he invested x dollars in the account that paid 9% interest, and (10,000 - x) dollars in the account that paid 7% interest.

The formula for calculating simple interest is: interest = principal * rate * time. In this case, the time is one year.

For the account that paid 9% interest, the interest earned would be: x * 0.09 * 1 = 0.09x.

For the account that paid 7% interest, the interest earned would be: (10,000 - x) * 0.07 * 1 = 0.07(10,000 - x).

According to the information given, the total interest earned is $740. So we can set up the equation: 0.09x + 0.07(10,000 - x) = 740.

Now, let's solve this equation:

0.09x + 0.07(10,000 - x) = 740
0.09x + 700 - 0.07x = 740
0.02x + 700 = 740
0.02x = 40
x = 40 / 0.02
x = 2,000

Juan invested $2,000 in the account that paid 9% interest. To find out how much he invested in the account that paid 7% interest, we subtract $2,000 from the total investment of $10,000:

10,000 - 2,000 = 8,000

Juan invested $2,000 in the account that paid 9% interest and $8,000 in the account that paid 7% interest.

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Zoe covers varying distances during her journey from her house to the bus stop. She starts from her house, covering 0 yards initially. As she walks towards the bus stop, the distance covered gradually increases, reaching a total of 460 yards when she arrives at the bus stop.

Zoe walks from her house to a bus stop that is 460 yards away. Let's explore the varying distances she would cover during different stages of her journey.

Stage 1: Zoe starts from her house.

At the beginning of her journey, Zoe is at her house. The distance covered at this stage is 0 yards since she hasn't started walking yet.

Stage 2: Zoe walks towards the bus stop.

Zoe starts walking from her house towards the bus stop, which is 460 yards away. As she progresses, the distance covered gradually increases. We can consider various checkpoints to track her progress:

- After walking for 100 yards, Zoe has covered a distance of 100 yards.

- After walking for 200 yards, Zoe has covered a distance of 200 yards.

- After walking for 300 yards, Zoe has covered a distance of 300 yards.

- After walking for 400 yards, Zoe has covered a distance of 400 yards.

- Finally, after walking for 460 yards, Zoe reaches the bus stop. The distance covered at this stage is the total distance from her house to the bus stop, which is 460 yards.

In summary, Zoe covers varying distances during her journey from her house to the bus stop. She starts from her house, covering 0 yards initially. As she walks towards the bus stop, the distance covered gradually increases, reaching a total of 460 yards when she arrives at the bus stop.

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After evaluation when y = 9, the value of -147 + 5₁ is -102.

Evaluation refers to the process of finding the value or result of a mathematical expression or equation. It involves substituting given values or variables into the expression and performing the necessary operations to obtain a numerical or simplified value. The result obtained after substituting the values is the evaluation of the expression.

To evaluate the expression -147 + 5₁ when y = 9, we substitute the value of y into the expression:

-147 + 5 * 9

Simplifying the multiplication:

-147 + 45

Performing the addition:

-102

Therefore, when y = 9, the value of -147 + 5₁ is -102.

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To determine the value of a, we consider the remainders obtained when 42 is divided by 13. The remainder of this division is 3, as 42 = 13 * 3 + 3.

To find the value of a, we start by simplifying the expression on the left-hand side of the congruence. By calculating 6^0+6 = 7, we have 6(7) = 42.

Next, we apply the congruence relation, a (mod 13), which means finding the remainder when a is divided by 13. In this case, we want to find the value of a that is congruent to 42 modulo 13.

To determine the value of a, we consider the remainders obtained when 42 is divided by 13. The remainder of this division is 3, as 42 = 13 x3 + 3.

Since the condition states that 0 ≤ a ≤ 12, we check if the remainder 3 falls within this range. As it does, we conclude that the value of a satisfying the given condition is a = 3.

Therefore, the value of a such that 0 ≤ a ≤ 12 and 6 (6⁰+6) = a (mod 13) is a = 3.

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Answer:

B. 7.5

Step-by-step explanation:

One right triangle is formed by:

Another similar right triangle is formed by:

Proportionality of similar sides:

First set of similar sides:

Second set of similar sides:

Now we can create proportions to solve for s, the length of the shadow:

18 / 6 = (15 + s) / s

(3 = (15 + s) / s) * s

(3s = 15 + s) - s

(2s = 15) / 2

s = 7.5

Thus, the length of the shadow is 7.5 ft.

Check the validity of the answer:

We can check our answer by substituting 7.5 for s and seeing if we get the same answer on both sides of the equation we just used to solve for s:

18 / 6 = (15 + 7.5) / 7.5

3 = 22.5 / 7.5

3 = 3

Thus, our answer is correct.

Answer:

B.  7.5

[tex]\hrulefill[/tex]

Step-by-step explanation:

The given diagram shows two similar right triangles.

Let "x" be the base of the smaller triangle. Therefore:

In similar triangles, corresponding sides are always in the same ratio. Therefore, we can set up the following ratio of base to height:

[tex]\begin{aligned}\sf \underline{Smaller\;triangle}\; &\;\;\;\;\;\sf \underline{Larger\;triangle}\\\\\sf base:height&=\sf base:height\\\\x:6&=(15+x):18\end{aligned}[/tex]

Express the ratios as fractions:

[tex]\dfrac{x}{6}=\dfrac{(15+x)}{18}[/tex]

Cross multiply and solve for x:

[tex]\begin{aligned}18x&=6(15+x)\\\\18x&=90+6x\\\\18x-6x&=90+6x-6x\\\\12x&=90\\\\\dfrac{12x}{12}&=\dfrac{90}{12}\\\\x&=7.5\end{aligned}[/tex]

Therefore, the shadow of the tree is 7.5 feet long.

(a) To find a basis for the row space of matrix A, we row-reduce the matrix to its row-echelon form and identify the linearly independent rows. The basis for the row space of A is {[-1, 0, 1], [0, 2, 8]}.

(b) To find a basis for the column space of matrix A, we identify the pivot columns from the row-echelon form of A. The basis for the column space of A is {[-1, -1], [0, 2], [1, 7]}.

(c) To find a basis for the null space of matrix A, we solve the homogeneous linear system A*u = 0 by row-reducing the augmented matrix. The basis for the null space of A is {[1, -4, 2]}.

(d) To determine if a vector ū is in the row space of A, we check if it is a linear combination of the basis vectors of the row space. ū = [1, 1, 1] is not in the row space, while ū = [2, 1, 1] is in the row space.

(e) To determine if vectors a = [1, 1] and b = [1, 5] are in the column space of A, we check if they are linear combinations of the basis vectors of the column space. Neither a nor b is in the column space of A.

(a) To find a basis for the row space of matrix A, we need to find the linearly independent rows of A.

Row-reduce the matrix A to its row-echelon form:

-1  0  1

-1  2  7

Perform row operations to simplify the matrix:

R2 = R2 + R1

-1  0  1

0  2  8

Now, we can see that the first row and second row are linearly independent. Therefore, a basis for the row space of matrix A is:

{[-1, 0, 1], [0, 2, 8]}

(b) To find a basis for the column space of matrix A, we need to find the linearly independent columns of A.

From the row-echelon form of A, we can see that the first and third columns are pivot columns. Therefore, a basis for the column space of matrix A is:

{[-1, -1], [0, 2], [1, 7]}

(c) To find a basis for the null space of matrix A, we need to solve the homogeneous linear system A*u = 0.

Setting up the augmented matrix:

-1  0  1 | 0

-1  2  7 | 0

Perform row operations to solve the system:

R2 = R2 + R1

-1  0  1 | 0

0  2  8 | 0

The row-echelon form of the augmented matrix suggests that the variable x and z are free variables, while the variable y is a pivot variable. Therefore, a basis for the null space of matrix A is:

{[1, -4, 2]}

(d) To determine if the vector ū = [1, 1, 1] is in the row space of A, we can check if ū is a linear combination of the basis vectors of the row space of A.

Since ū is not a linear combination of the basis vectors [-1, 0, 1] and [0, 2, 8], it is not in the row space of A.

To determine if the vector ū = [2, 1, 1] is in the row space of A, we follow the same process. Since ū is a linear combination of the basis vectors [-1, 0, 1] and [0, 2, 8] (2 * [-1, 0, 1] + [-1, 2, 7] = [2, 1, 1]), it is in the row space of A.

(e) To determine if the vectors a = [1, 1] and b = [1, 5] are in the column space of matrix A, we can check if they are linear combinations of the basis vectors of the column space of A.

The column space of matrix A is spanned by the vectors [-1, -1], [0, 2], and [1, 7].

For vector a = [1, 1]:

1 * [-1, -1] + 0 * [0, 2] + 1 * [1, 7] = [0, 6]

Since [0, 6] is not equal to [1, 1], vector a is not in the column space of A.

For vector b = [1, 5]:

1 * [-1, -1] + 2 * [0, 2] + 0 * [1, 7] = [-

1, 9]

Since [-1, 9] is not equal to [1, 5], vector b is not in the column space of A.

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(a) The reading on the thermometer will be 7°C after 2 more minutes.

(b) The thermometer will read 2°C 15 minutes after it was taken outdoors.

(a) In the given scenario, the temperature on the thermometer decreases by 8°C in the first minute (from 22°C to 14°C). We can observe that the temperature change is linear, decreasing by 8°C per minute. Therefore, after 2 more minutes, the temperature will decrease by another 2 times 8°C, resulting in a reading of 14°C - 2 times 8°C = 14°C - 16°C = 7°C.

(b) To determine when the thermometer will read 2°C, we need to find the number of minutes it takes for the temperature to decrease by 20°C (from 22°C to 2°C). Since the temperature decreases by 8°C per minute, we divide 20°C by 8°C per minute, which gives us 2.5 minutes. However, since the thermometer cannot read fractional minutes, we round up to the nearest whole minute. Therefore, the thermometer will read 2°C approximately 3 minutes after it was taken outdoors.

It's important to note that these calculations assume a consistent linear rate of temperature change. In reality, temperature changes may not always follow a perfectly linear pattern, and various factors can affect the rate of temperature change.

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The bearing of the line between points A and C is S 46°40'47" E.

To determine the bearing of the line between points A and C, we need to calculate the relative angle between the backsight bearing from point A to point B and the angle measured right to point C.

The backsight bearing from point A to point B is given as S 89°54'59" E.

The angle measured right to point C is given as 136°14'12".

To calculate the bearing of the line between points A and C, we need to subtract the angle measured right from the backsight bearing.

Since the backsight bearing is eastward (E) and the angle measured right is clockwise, we subtract the angle from the backsight bearing.

Therefore, the bearing of the line between points A and C is S 46°40'47" E.

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1.1.1 x² - y ≥ 0 ⇒ y ≤ x². This means that the domain of the function is the set of all points (x, y) such that y ≤ x². The domain of the function is therefore D = {(x, y) : y ≤ x²}.

The domain of a function is defined as the set of all possible values of the independent variable for which the function is defined.

To find the domain of the function f(x, y) = 1/√(x² - y), we need to make sure that the radicand is not negative. As a result, x² - y ≥ 0 ⇒ y ≤ x². This indicates that the set of all points (x, y) such that y x2 is the function's domain.

Therefore, the function's domain is D = " {(x, y) : y ≤ x²}.."

1.1.2 To find the range of the function, we can start by looking at the behavior of the function as x tends to infinity and negative infinity. As x → ±∞, the denominator of the function approaches infinity, and therefore the function approaches zero. The function is also defined only for non-negative values of x since the argument of the radical must be non-negative. Since we can make the function as small as we want, but never negative, the range of the function is the set of all non-negative real numbers.

Range of the function f(x,y) = 1/√(x² - y) is given by R = [0, ∞).

1.2 To sketch the level curves of the function f(x, y) = 4x² + 9y² at f = 1/2, 1, and 2, we need to solve the equation 4x² + 9y² = k for each value of k and sketch the curve that corresponds to the solution.

1.2.1 At f = 1/2, we have 4x² + 9y² = 1/2. Rearranging, we get y²/(1/8) + x²/(1/2) = 1. This is the equation of an ellipse with semi-major axis a = √2 and semi-minor axis b = 1/√2. The center of the ellipse is at the origin, and the ellipse lies in the first and third quadrants.

1.2.2 At f = 1, we have 4x² + 9y² = 1. Rearranging, we get y²/(1/9) + x²/(1/4) = 1. This is the equation of an ellipse with semi-major axis a = 3/2 and semi-minor axis b = 1/2. The center of the ellipse is at the origin, and the ellipse lies in the first and third quadrants.

1.2.3 At f = 2, we have 4x² + 9y² = 2. Rearranging, we get y²/(2/9) + x²/(1/2) = 1. This is the equation of an ellipse with semi-major axis a = 3 and semi-minor axis b = 3/√2. The center of the ellipse is at the origin, and the ellipse lies in the first and third quadrants.

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In a normal probability distribution, normal curve is symmetric about: mean. The Option C.

In a normal probability distribution, the normal curve is symmetric about the mean. This means that the curve is equally balanced on both sides of the mean, creating a mirror image.

The mean represents the center or average value of the distribution, and the symmetry indicates that the probabilities of observing values to the left and right of the mean are equal. The standard deviation and variance play important roles in describing the spread or dispersion of the distribution, but they do not determine the symmetry of the curve.

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The correct answer is c. mean. The normal curve is symmetric about the mean.

In a normal probability distribution, the normal curve is symmetric about the mean. This fundamental property of the normal distribution is one of its defining characteristics. It means that the probability density function of a normal distribution is perfectly symmetrical, with the highest point of the curve located at the mean.

The mean is the central value of a normal distribution and represents its location or center point. The symmetric nature of the normal curve implies that the probabilities of observing values to the left and right of the mean are equal. This symmetry indicates that the mean, as well as the median and mode, are all located at the same point on the distribution.

On the other hand, the variance and standard deviation are measures of dispersion or spread within the distribution. They quantify how data points deviate from the mean. While the variance and standard deviation are important characteristics of a normal distribution, they do not affect the symmetry of the normal curve.

Therefore, the correct answer is c. mean. The normal curve is symmetric about the mean.

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The 10 kg child should sit 0.6 meters from the axis of rotation on the seesaw to achieve equilibrium.

To achieve equilibrium on the seesaw, the total torque on each side of the seesaw must be equal. Torque is calculated by multiplying the weight (mass x gravity) by the distance from the axis of rotation.

Let's calculate the torque on each side of the seesaw: -

Child weighing 18 kg:

torque = (18 kg) x (9.8 m/s²) x (2 m)

           = 352.8 Nm

Child weighing 21 kg:

torque = (21 kg) x (9.8 m/s²) x (2 m)

           = 411.6 Nm

To find the position where a 10 kg child should sit to achieve equilibrium, we need to balance the torques.

Since the total torque on one side is greater than the other, the 10 kg child needs to be placed on the side with the lower torque.

Let x be the distance from the axis of rotation where the 10 kg child should sit. The torque exerted by the 10 kg child is:

(10 kg) x (9.8 m/s^2) x (x m) = 98x Nm

Equating the torques:

352.8 Nm + 98x Nm = 411.6 Nm

Simplifying the equation:

98x Nm = 58.8 Nm x = 0.6 m

Therefore, to attain equilibrium, the 10 kg youngster should sit 0.6 metres from the seesaw's axis of rotation.

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The angle of depression from the person is approximately 20.2 degrees.

To find the angle of depression, we can consider the triangle formed by the person, the boat, and the vertical line from the person to the water surface. The person is 60 feet above the water, and the cable connecting them to the boat is 170 feet long.

The angle of depression is the angle formed between the cable and the horizontal line. This angle can be found using trigonometry. We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side.

In this case, the opposite side is the height of the person (60 feet) and the adjacent side is the horizontal distance between the person and the boat. Let's denote this distance as x.

Using the tangent function, we have:

tan(angle) = opposite / adjacent

tan(angle) = 60 / x

To find the value of x, we can use the Pythagorean theorem, which states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. In this case, the hypotenuse is the length of the cable (170 feet), and the legs are the height of the person (60 feet) and the horizontal distance (x).

Applying the Pythagorean theorem, we have:

x^2 + 60^2 = 170^2

x^2 + 3600 = 28900

x^2 = 28900 - 3600

x^2 = 25300

x = √25300

x ≈ 159.1 feet

Now, we can substitute the value of x into the tangent equation to find the angle:

tan(angle) = 60 / 159.1

Using a calculator, we can calculate the inverse tangent (arctan) of this ratio:

angle ≈ arctan(60 / 159.1)

angle ≈ 20.2 degrees

As a result, the angle of depression with respect to the person is roughly 20.2 degrees.

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A basis for S is { [3 + 6x₃, 1, x₃, 0], [6x₃ - 6, 0, x₃, 1] }, where x₃ is a free variable.

To find a basis for the subspace S consisting of the solutions to the given system of equations, we can first express the system in matrix form:

A * X = 0

Where A is the coefficient matrix and X is the vector of variables:

A = | 0 4 8 -4 |

| 1 -3 -6 6 |

| 0 -3 -6 3 |

To find the basis for S, we need to find the solutions to the homogeneous system A * X = 0. We can do this by finding the row echelon form (REF) of the augmented matrix [A | 0] and identifying the free variables.

Performing row operations, we obtain the REF:

| 1 -3 -6 6 |

| 0 4 8 -4 |

| 0 0 0 0 |

From the REF, we can see that the third column of A is a pivot column, while the second and fourth columns correspond to the free variables. Let's denote the free variables as x₂ and x₄.

To find a basis for S, we can set x₂ = 1 and x₄ = 0, and solve for the other variables:

x₁ - 3(1) - 6x₃ + 6(0) = 0

x₁ - 3 - 6x₃ = 0

x₁ = 3 + 6x₃

Therefore, a possible solution is X = [3 + 6x₃, 1, x₃, 0].

Similarly, setting x₂ = 0 and x₄ = 1, we have:

x₁ - 3(0) - 6x₃ + 6(1) = 0

x₁ - 6x₃ + 6 = 0

x₁ = 6x₃ - 6

Another possible solution is X = [6x₃ - 6, 0, x₃, 1].

Hence, a basis for S is { [3 + 6x₃, 1, x₃, 0], [6x₃ - 6, 0, x₃, 1] }, where x₃ is a free variable.

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