Applications of Differential Equations : Mass-spring system


The motion of a certain mass in a spring is described by:

[tex]\sf x''+2x'+2x=5sin(t)[/tex]


a) What is the amplitude of the resulting constant periodic oscillations?


b) Suppose the mass is initially at rest, that is, x(0)=0 and x′(0)=0, find the position function x(t).

Answer:

Hello, so most of your steps are in the correct order. All you need to do is switch is the following steps.

x = -3 ± 5/2 should go before x = -3 + 5/2 and x = -3 - 5/2 as the second seperates the the ± into + and -

[You need to switch steps 4 and 5]

Answer:

5x - 9

Step-by-step explanation:

2x -(-3x + 9)                

2x + 3x - 9)                  DISTRIBUTE -1

5x - 9                           COMBINE LIKE TERMS

The function y = 18(0.914)^m is having exponential decay. The decay percent of this function is 8.6%

Consider the function:

[tex]y = a(1\pm r)^m[/tex]

where m is the number of times this growth/decay occurs, a = initial amount, and r = fraction by which this growth/decay occurs.

If there is plus sign, then there is exponential growth happening by r fraction or 100r %

If there is negative sign, then there is exponential decay happening by r fraction or 100r %

For this case, the exponential growth/decay function given is:

[tex]y = 18(0.914)^m[/tex]

That means, there is multiplication of (0.914) for each growth/decay.

We can rewrite 0.914 as:

0.914 = 1 - 0.086

This shows decay as multiplication with 0 < quantity < 1 makes the number less in magnitude.

A quantity times (1 - 0.086) = that quantity - that quantity times 0.086

and we can write 0.086 = 8.6/100 = 8.6%

Thus, we're decreasing 8.6% each time from a quantity when we multiply that quantity with (1-0.086)

Thus, there is exponential(exponential because of the presence of variable 'm' as exponent in the given function) decay of 8.6%.

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Answer:

25

Step-by-step explanation:

The formula for the Pythagorean Theorem is:

[tex]\sqrt{a^2+b^2}=c[/tex]

Now,

A=7

B=24

7x7=49

24x24=576

Add,

576+49=625

Last,

Take the square root of 625

The square root of 625 is 25

c=25

To Check if the hypotenuse is correct:

[tex]a^2+b^2=c^2[/tex]

[tex]7^2+24^2=25^2[/tex]

7x7=49

24x24=576

25x25=625

49+576=625

625=625,

Therefore the hypotenuse is 25.

Answer:

80 pages

Step-by-step explanation:

Divide 60 by 3 to get 20 because this is how many pages he can read in 1/4 hour. Then multiply that by 4 to get 80, which is how many pages he can read in a full hour.

Answer:

answer in order from top to bottom

1. True

2. false

3. true

4. false

Answer:

.

Step-by-step explanation:

1.true

2.true

3.false

4.false

i hope it helps :)

Answer:

D is the correct answer

Step-by-step explanation:

A is not correct because it does not meet the fourth criteria, B is not correct because it doesnt meet the second criteria, C is not correct because it does not meet the third criteria, D IS CORRECT because it meets the fourth criteria.

Answer:

The answer is

→ No property

Step-by-step explanation:

Given equation: [tex]2x + 9 = 4x + 15[/tex]

What were supposed to do: Which properly justifies step 4.

Solve:

[tex]2x + 9 = 4x + 15[/tex]

Step 1: We must subtract 4x on both sides.

[tex]2x+9-4x=4x+15-4x[/tex]

[tex]= -2x+9=15[/tex]

Step 2: We must subtract 9 on both sides.

[tex]-2x+9-9=15-9[/tex]

[tex]= -2x=6[/tex]

Step 3: We must divide both sides by -2.

[tex]\frac{-2}{2} = \frac{6}{-2}[/tex]

[tex]= x = -3[/tex]

Result:

[tex]x=-3[/tex]

There is no Step 4.

Answer:

42.5 ft squared

Step-by-step explanation:

hope this helped brah

Answer:

B. Bangladesh

Step-by-step explanation:

he UN has called for Burma to allow access to a Fact Finding Mission to investigate reports of human rights violations and abuses and to work with Bangladesh to facilitate repatriation of Rohingya refugees, and in September 2018 the International Criminal Court (ICC) determined it had jurisdiction to investigate reported human rights abuses against Rohingyas. Burma has rejected charges of ethnic cleansing and genocide, and has chosen not to work with the UN Fact Finding Mission or the ICC.

1) K=17°

2)Jk=2.84yd

3)KL=2.71yd

Answer:

I'm assuming where you wrote x2, you meant 2x.

Anyway, if csc (x) = 2, since csc(x) = 1/sin(x), we know that 1/sin x = 2, so sin (x) = 1/2. Since sin2x + cos2x = 1, cos2x = 1 - sin2x, so cos x = √1 - sin2x. In this case, cos x = √ 1 - (1/2)2 = √(3/4) = (√3)/2. Finally, since sin x = opp / hyp , cos = adj/ hyp: sin x / cos x = (opp/hyp)/(adj/hyp) = opp/adj = tan x. So this means that tan x = sin x / cos x = 1/2 / √ 3 / 2 = 1/√3 = √3 / 3

At this point, all that is necessary is to use the double angle formulas for sin, cos, and tan.

sin (2x) = 2 sin x cos x = 2 (1/2) (√3/2) = √3 / 2

cos (2x) = cos2x - sin2x = (√3/2)2 - (1/2)2 = 3/4 - 1/4 = 1/2

tan (2x) = 2 tan x / (1 - tan2x) = 2 (√3 / 3) / ( 1 - (√3/3)2) = 2/√3 / (1 - 1/3) = 2/√3 / (2/3) = 3/√3 = √3

To sum up, sin (2x) = √4 / 2, cos (2x) = 1/2, and tan(2x) = ✓3.

Step-by-step explanation:

BRAIN LY FAST

MARK A BRAINLESS

[tex]\sin\left(\frac{x}{2}\right) = \pm\frac{\sqrt{4 + \sqrt{15}}}{\sqrt{8}}\\\cos\left(\frac{x}{2}\right) = \pm\frac{\sqrt{4 - \sqrt{15}}}{\sqrt{8}}\\\tan\left(\frac{x}{2}\right) = \pm\frac{\sqrt{4 + \sqrt{15}}}{\sqrt{4 - \sqrt{15}}}\\[/tex]

To find sin(x/2), cos(x/2), and tan(x/2), we need to use the half-angle identities for trigonometric functions.

Given: csc(x) = 4 for 90° < x < 180°

Step 1: Find sin(x). Since csc(x) is the reciprocal of sin(x), we have:

csc(x) = 1/sin(x) = 4

Now, find sin(x):

sin(x) = 1/csc(x) = 1/4

Step 2: Find cos(x). To do this, use the Pythagorean identity:

[tex]sin^2(x) + cos^2(x) = 1\\cos^2(x) = 1 - sin^2(x) \\= 1 - (1/4)^2 \\= 1 - 1/16 \\= 15/16[/tex]

Now, find cos(x):

[tex]\cos(x) = \pm \sqrt{\frac{15}{16}}[/tex]

Since x is in the second quadrant (90° < x < 180°), cos(x) is negative. So,

[tex]\cos(x) = - \sqrt{\frac{15}{16}}[/tex]

[tex]= -\sqrt{15} / 4[/tex]

Step 3: Find tan(x). To do this, use the identity:

[tex]\tan(x) = \frac{\sin(x)}{\cos(x)}\\\tan(x) = \frac{\frac{1}{4}}{-\frac{\sqrt{15}}{4}}\\\tan(x) = -\frac{1}{\sqrt{15}}[/tex]

Now, let's find sin(x/2), cos(x/2), and tan(x/2) using the half-angle identities:

Half-angle identities:

[tex]\sin \left ( \frac{x}{2} \right ) = \pm \sqrt{\frac{1 - \cos x}{2}}\\\cos \left ( \frac{x}{2} \right ) = \pm \sqrt{\frac{1 + \cos x}{2}}\\\tan \left ( \frac{x}{2} \right ) = \frac{\sin \left ( \frac{x}{2} \right )}{\cos \left ( \frac{x}{2} \right )}[/tex]

Step 4: Find sin(x/2):

[tex]\sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 - \cos x}{2}}\\\sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 + \frac{\sqrt{15}}{4}}{2}}\\\sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{4 + \sqrt{15}}{8}}\\\sin \left( \frac{x}{2} \right) = \pm \frac{\sqrt{4 + \sqrt{15}}}{\sqrt{8}}\\\sin \left( \frac{x}{2} \right) = \pm \frac{\sqrt{4 + \sqrt{15}}}{2}[/tex]

Step 5: Find cos(x/2):

[tex]\cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 + \cos(x)}{2}}\\\cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 + \left(-\frac{\sqrt{15}}{4}\right)}{2}}\\\cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 - \frac{\sqrt{15}}{4}}{2}}\\\cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{4 - \sqrt{15}}{8}}\\\cos\left(\frac{x}{2}\right) = \pm\frac{\sqrt{4 - \sqrt{15}}}{\sqrt{8}}\cos\left(\frac{x}{2}\right) = \pm\left(\frac{\sqrt{4 - \sqrt{15}}}{\sqrt{8}}\right)[/tex]

Step 6: Find tan(x/2):

[tex]\tan \left ( \frac{x}{2} \right ) = \frac{\sin \left ( \frac{x}{2} \right )}{\cos \left ( \frac{x}{2} \right )} = \pm \frac{\sqrt{4 + \sqrt{15}}}{\sqrt{4 - \sqrt{15}}}[/tex]

So, the values are:

[tex]\sin\left(\frac{x}{2}\right) = \pm\frac{\sqrt{4 + \sqrt{15}}}{\sqrt{8}}\\\cos\left(\frac{x}{2}\right) = \pm\frac{\sqrt{4 - \sqrt{15}}}{\sqrt{8}}\\\tan\left(\frac{x}{2}\right) = \pm\frac{\sqrt{4 + \sqrt{15}}}{\sqrt{4 - \sqrt{15}}}\\[/tex]

Note: The ± symbol indicates that there are two possible solutions due to the quadrant restrictions.

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Complete Question is:

If csc ( x ) = 4, for 90 ∘ < x < 180, then sin ( x /2 ) = ? cos ( x /2 ) = ? tan ( x /2 ) = ?

Answer:

70% to about 85% of your maximum heart rate

Step-by-step explanation:

Answer:

He would save $19.20.

Step-by-step explanation:

30% = 0.3

64*0.3 = 19.2

There he saves $19.20 on the coat.

Answer:

b=-30;c=41

Step-by-step explanation:

I don't really know how to explain it... Vertex form, so factor out 5, divide by two, that has to be three. so .1b=3, b=30... opposite of the square of 3 times 5 add c=-4, so -45+c=-4, c=41. Just learn vertex form, and complete the perfect square. Good luck!

Answer:

35 ft²

Step-by-step explanation:

The shape is a triangle and its area is given by the formula below:

Area of triangle= ½ ×base ×height

Do note that the height has to be the perpendicular height, which is 5 ft in this case.

Area of triangle

= ½(14)(5)

= 35 ft²

Answer:

40

Step-by-step explanation:

16+24=40

40-16=24

Answer:

a = 40

Step-by-step explanation:

Add 16 on both sides

a - 16 + 16 = 24 + 16

a = 24 +16 = 40

a = 40

Verification

LHS = a - 16  a = 40

40 - 16 = 24

RHS = 24

Both LHS and RHS are same so, our answer is correct.

Answer:

c+3<22

Step-by-step explanation:

c+3<22

Answer:Im pretty sure the slope is 2.

Step-by-step explanation:

The lines are

i)  y=-x+6

ii) y=2x-3

The solution of the system of equations is found by equalizing the 2 equations:

-x+6=2x-3

-2x-x=-6-3

-3x=-9

x=-9/(-3)=3

substitute x=3 in either i) or ii):

i)  y=-3+6=3

ii) y=2(3)-3=6-3=3

(the result is the same, so checking one is enough)

This means that the point (3, 3) is a point which is in both lines, so a solution to the system.

In graphs, this means that the lines intersect at (3, 3) ONLY

Answer: The graph where the lines intersect at (3, 3)

Answer:In the following diagram, two parallel lines are cut by a transversal. What is the value of x x? Solution: The two angles 3x −15 3 x − 15 and 2x +7 2 x + 7 are equivalent. That is: 3x −15 = 2x +7 3 x − 15 = 2 x + 7

In the following diagram, two parallel lines are cut by a transversal. What is the value of x x? Solution: The two angles 3x −15 3 x − 15 and 2x +7 2 x + 7 are equivalent. That is: 3x −15 = 2x +7 3 x − 15 = 2 x + 7

Step-by-step explanation: : 3

Answer:

x= 29

Step-by-step explanation:

use Pythagoras because it has a right angle and 2 sides length is given.

when the hypoteneus(side opposite the right angle) is the value you are finding out you use the equation : x²+y²=hypoteneus² (Pythagoras)

then you sub in the values you are given

(21)² + (20)² = hypoteneus²

them take the square root (²) to the opposite side to find the hypoteneus

square root (441+400) = hypoteneus

now you can put into a calculator and you get the value

hypoteneus =29 (for reference to hypoteneus is the same as X-the value you are looking for)

therefore, x=29

for extra reference, say you want to work out a side that is not the hypotenues you use the formula ,

(side u looking for)² = (hypotenues)² - (other side given)²

Answer:

x = 29

Step-by-step explanation:

To find x, we can use the Pythagorean theorem.

a^2 + b^2 = c^2

21^2 + 20^2 = x^2

441 + 400 = x^2

841 = x^2

[tex]\sqrt{841}[/tex] = [tex]\sqrt{x^2}[/tex]

29 = x

Answer:

6 and -1

Step-by-step explanation:

If you take the equation and set it to be 0, you can find the input values that make it 0.

[tex]x^2-5x-6=0[/tex]

Since this is a quadratic, equation you can use the quadratic formula with the values:

The quadratic formula is:

[tex]x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]

If you plug the values for a, b, and c, you get:

x = 6 and x = -1

Answer:

Step-by-step explanation:

Beth's position over time is constant, which makes the problem a lot easier. You are asked for the rate of position versus time, so begin by doing that. Since the rate is constant, clearly from the table, you can take any two data points and take their ratio. Her rate would be 11 feet every 5 seconds, or equivalently 2.2 feet every second. To graph the data, put position on the x-axis and time on the y-axis. The slope will be 2.2 at every point along her bike ride.

Moving on to the third part. If Amy traveled at a rate of 75 feet every 30 seconds, then she would be traveling 2.5 feet every second. Amy's rate is larger than Beth's rate. Since the rate corresponds to the slope of the postion vs. time graph, Amy's graph will have a steeper incline since the slope is 2.5

Answer:

Inf solutions.

Step-by-step explanation:

1/2 (b+14) = b+14/2

Simplify

b+14 = b+14

B=b

Since the answers are equal on both sides there are infinite solutions,

Answer:

4. 3 cans of paint will cover 60% of Dalia's room

5. Dalia's entire room requires 5 cans of paint

Step-by-step explanation:

If Dalia has painted 1/10 of her room with 1/2 can of paint, then she will paint 2/10 of her room with 1 can of paint (since there are 2 halves in a whole).

2/10 = 1/5  ⇒  1 can of paint covers 1/5 of the room.

Therefore, to paint the entire room, she will need 5 cans of paint.

60% of 5 cans of paint = 0.6 × 5 = 3

Therefore, 3 cans of paint will cover 60% of the room.

Answer:

7

Step-by-step explanation:

apply the Pythagorean theorem

[tex]10 {}^{2} = ( \sqrt{51} ) {}^{2} + x {}^{2} \\ 100 = 51 + x {}^{2} \\ 100 - 51 = x {}^{2} \\ 49 = x {}^{2} \\ x = 7[/tex]