A ball is dropped from a height of 9 m. Each time it strikes the ground it bounces vertically to a height that is 2/3 of the preceding height. Find the total distance the ball will travel if it is assumed to bounce infinitely often. show all work.

[tex]\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &250\\ r=rate\to 16\%\to \frac{16}{100}\dotfill &0.16\\ t=\textit{elapsed time} \end{cases} \\\\\\ A=250(1 + 0.16)^{t}\implies A=250(1.16)^t \\\\[-0.35em] ~\dotfill[/tex]

[tex]A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\dotfill &1000\\ P=\textit{initial amount}\dotfill &250\\ r=rate\to 16\%\to \frac{16}{100}\dotfill &0.16\\ t=\textit{elapsed time} \end{cases} \\\\\\ 1000=250(1.16)^t\implies \cfrac{1000}{250}=1.16^t\implies 4=1.16^t \\\\\\ \log(4)=\log(1.16^t)\implies \log(4)=t\log(1.16) \\\\\\ \cfrac{\log(4)}{\log(1.16)}=t\implies \stackrel{\textit{about 9 years and 4 months}}{9.34\approx t}[/tex]

Answer:

[tex]\boxed{\text{Slope = 5}}[/tex]

[tex]\boxed{\text{y-intercept = 1}}[/tex]

Step-by-step explanation:

To find the slope of the line, we must pick any two points on the line. Using those two points, we will calculate the Rise and the Run. Then, we will use the slope formula (Rise/Run) to find the slope. The y-intercept is the intersection of the point on the y-axis.

[tex]\text{My chosen points: (0,1) and (1,6)}[/tex]

[tex]\text{Rise = 5; Run = 1}[/tex]

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[tex]\text{Slope}= \dfrac{\text{Rise}}{\text{Run}}[/tex]

➡ [tex]\bold{Slope = \frac{5}{1} = \underline{5}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [\text{Rise = 5; Run = 1}][/tex]

[tex]\underline{........................................................................................................................}[/tex]

[tex]\text{Intersection on y-axis:} \ (0,1)[/tex]

➡ [tex]\text{y-intercept} \rightarrow (0,\bold{ 1}) \rightarrow \bold{\underline{{1}}}}[/tex]

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The Laplace transform of the non-homogeneous second order differential equation is [tex]\mathcal {L} \{f(t)\} = \frac{4\cdot (s+6)}{s\cdot (s-4)\cdot (s+1)\cdot (s-5)} +\frac{1}{s} -\frac{1}{s\cdot (s-5)}[/tex].

A Laplace transform is a frequency-based algebraic substitution method used to determine the solutions of differential equations in a quick and efficient manner.

In this question we shall use the following Laplace transforms:

[tex]\mathcal {L} \{f(t) + g(t)\} = \mathcal {L} \{f(t)\} + \mathcal {L}\{g(t)\}[/tex]   (1)

[tex]\mathcal {L} \{\alpha\cdot f(t)\} = \alpha\cdot \mathcal {L} \{f(t)\}[/tex]   (2)

[tex]\mathcal{L} \left\{y^{(n)} \right\} = s^{n}\cdot \matcal {L}\{f(t)\}-s^{n-1}\cdot y(0) -...-y^{(n)}(0)[/tex]   (3)

[tex]\mathcal {L} \{e^{-a\cdot t}\} = \frac{1}{s+a}[/tex]   (4)

Now we proceed to derive an expression fo the Laplace transform of the solution of the differential equation:

[tex]y'' -5\cdot y' = 8\cdot e^{4\cdot t}-4\cdot e^{-t}[/tex]

[tex]s^{2}\cdot \mathcal {L}\{f(t)\}-5\cdot y(0) - y'(0) - 5\cdot s \cdot \mathcal {L} \{f(t)\} +5\cdot y(0) = \frac{8}{s-4}-\frac{4}{s+1}[/tex]

[tex]\mathcal {L} \{f(t)\} = \frac{4\cdot (s+6)}{s\cdot (s-4)\cdot (s+1)\cdot (s-5)} +\frac{1}{s} -\frac{1}{s\cdot (s-5)}[/tex]

The Laplace transform of the non-homogeneous second order differential equation is [tex]\mathcal {L} \{f(t)\} = \frac{4\cdot (s+6)}{s\cdot (s-4)\cdot (s+1)\cdot (s-5)} +\frac{1}{s} -\frac{1}{s\cdot (s-5)}[/tex]. [tex]\blacksquare[/tex]

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The Laplace transform Y'' − 5y' = 8e4t − 4e−t when y(0) = 1, y'(0) = −1 for f(t) = 8e4t − 4e−t would be  [tex]\frac{4(s+6)}{s.(s-4)(s+1).(s-5)} + \frac{1}{s} - \frac{1}{s.(s-5)}[/tex].

A Laplace transform is a frequency-based algebraic substitution method used to determine the solutions of differential equations quickly and efficiently.

The Laplace transform

L[ f(t) + g(t) ] = Lf(t) + Lg(t)

Also,

L [[tex]y^{n}[/tex]] = [tex]s^{n} . L[ f(t) ] - s^{n-1} .y(0) .....y^{n}(0)[/tex]

We have

Y'' − 5y' = 8 . e^4t − 4 . e−t

[tex]s^{2} . L[ f(t) ] - 5. y(0)- y'(0) .....y^{n}(0)[/tex]

y(0) = 1, y'(0) = −1  

for f(t) = 8e4t − 4e−t.

= [tex]\frac{8}{s-4} - \frac{4}{s+1}[/tex]

L [f(t)] = [tex]\frac{4(s+6)}{s.(s-4)(s+1).(s-5)} + \frac{1}{s} - \frac{1}{s.(s-5)}[/tex]

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Answer:

Step-by-step explanation:

The radious is 7 inches, that means the diameter is 14 inches.

The height is 18 inches. so...

V=πr2h=π·72·18≈2770.88472

So hence, your answer is

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Answer:

160 000

Step-by-step explanation:

400 000 - 240 000 = 160 000

Answer:

1. Find the circumference, then multiply that by 120 turns.

Formula for circumference;

C = 2πr

But, we need to find the radius which is 1/2(half) of the diameter, so,

D/2 = R

D = Diameter

R = Radius

20/2 = 10 is the radius.

Plug in values now:-

C = 2(3.14) x 10

C = 6.28 x 10

C = 62.8 is the circumference, now we multiply this by 120 to get the total amount of centimetres it'll go.

62.8(120)

= 7,536 cm it'll go.

2. Find the circumference, then multiply that by 750 turns/revolutions.

C = 2πr

Find radius,

D/2 = R

50/2 = 25 is the radius.

Plug in values now:-

C = 2(3.14) x 25

C = 6.28 x 25

C = 157 cm it'll go.

3. Same thing, find circumference then compare values.

Given that radius is 10,

Use the formula for circumference;

C = 2πr

Replace values:-

C = 2(3.14) x 10

C = 6.28 x 10

C = 62.8 cm is the circumference of the first circle.

Now the second circle, (use same formula)

C = 2πr

Find radius,

D/2 = R

10/2 = 5 is the radius.

Replace values:-

C = 2(3.14) x 5

C = 6.28 x 5

C = 31.4 cm is the circumference for the second circle.

Now subtract the second circle's circumference from the first:

62.8 - 31.4

= 31.4, 31.4 cm is how much greater the second circle is than the first.

4 - 5. Find circumference, (using circumference formula)

C = 2πr

Find radius,

D/2 = R

760/2 = 380

Replace values:-

C = 2(3.14) x 380

C = 6.28 x 380

C = 2,386.4 km is the distance he'll travel.

If he wants to travel around it with 2,000 km, he'll need to travel at least once.

Answer:

13.75 is your answer

Step-by-step explanation:

find area of whole triangle 5.5*7*1/2=19.25

Find area of blank spot 5.5*2*1/2=5.5

Subtract 19.25-5.5=13.75

Answer:

$1.5

Step-by-step explanation:

12 x 1.75 + 6c = 30

21 + 6c = 30

6c = 30 - 21

6c = 9

c = 9/6

c = 1.5

Answer:

do face reveal

Step-by-step explanation:

we don't talk about bruno

Answer:

yes

Step-by-step explanation:

Answer:

Delta does not have a relative extreme in the equation, so it is variable y It can not be true

Step-by-step explanation:

[tex]s = 5x + 3y[/tex]

There is an answer

The rest of the equation that has no substituted delta is not true, so the prime variables have relative extremes.

The function D represents the cost is  $175  for of x towels.

For the first 20 towels, the equation is simply 5x.

The 5 is the cost per towel ($5) times the number of towels sold up to and including 20 towels.

Replace x by 20 in given expression  we get the maximum cost

So the maximum cost of those 20 towels is $5 (20)=$100.

Looking at towels 21 and greater, the price drops to $3 each. Putting this in a formula,

Therefore we get,

3(x-20)+100

Which is $3 for the cost per towel, (x-20) since it starts with towel number 21, and +100 for the cost of the first 20 towels.

Let's try it for 45 towels using the formula

3 (45-20)+ 100

=3(25) + 100

=75+ 100

= $175

The function D represents the cost is  $175  for of x towels.

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Answer:

d

Step-by-step explanation:

edge 2022

Answer:

Step-by-step explanation:

29-9=20

9-(-11)=20

50×-20=-1000

-1000+29=-971

Answer:

c = 40degrees

Step-by-step explanation:

the sum of a triangle should be 180 degrees

you need to ad 80+60 which is 140 then minus 140 from 180 which is 40 so 40 degrees is the measurement of angle c

Answer:

0.1 or 10%

Step-by-step explanation:

Cost: 500

Sell: 550

550-500 = 50

50/500 = 0.1 OR 10%

Therefore, the percentage profit for each sale is 10% of the original wholesale price.

Answer:

10%

Step-by-step explanation:

profit=buyingpirce _selling price

550_500=50

%profit=profit\buyingprice×100

50\500×100=10%

Answer:

39

Step-by-step explanation:

you can set a proportion

84/156 = 21/x

when u solve for x u get 39

Answer:

78.42

Step-by-step explanation:

[tex]1.06(24.99 + 48.99)[/tex]

[tex]1.06 \times 73.98 = 78.42[/tex]

Answer:

15

Step-by-step explanation:

if 2x-10=12 then

2x=22 so

X=11

X+4= 15

Answer:

15

Step-by-step explanation:

1. we find what x is by finding out the first equation:

2x - 10 = 12

Add 10 to both sides (it cancels out on the first side)

-10 + 10 = 0 (cancels out)

12 + 10 = 22

2x = 22

Now we divide both sides by 2 to get x alone

22 divided by 2 = 11

X = 11

2. Now we can solve the equation x + 4:

Just fill in x, which as we solved, is 11

x + 4 = 11 + 4 = 15

Therefore, the value of x + 4 is 15.

The cosine function for the ferris wheel ride is an illustration of a sinusoidial function

The equation of the Ferris wheel is y = -190cos(π / 120 t) + 195

A cosine function is represented as:

y = Acos(Bt - C) + D

From the complete question, the diameter of the ferris wheel is 380 feet.

The amplitude represents the radius, and this is calculated as:

A = 380/2

A = 190

The function becomes:

y = 190cos(Bt - C) + D

The period of the function is:

T = 2π / B

From the complete question, one full rotation is completed in 4 minutes.

Convert the time to seconds

T = 4 * 60

T = 240.

So, we have:

240 = 2π / B

Divide both sides by 2

120 = π / B

Make B the subject

B = π / 120

The function becomes

y = 190cos(π / 120 t - C) + D

From the question, the ferris wheel is 195 feet above the ground.

This represents the vertical shift.

So, we have:

D = 195

The function becomes

y = 190cos(π / 120 t - C) + 195

Also, we have:

The lowest point is at t = 0 and the function is a negative cosine function

So, we have:

C = 0

The function becomes

y = -190cos(π / 120 t) + 195

Hence, the cosine function is y = -190cos(π / 120 t) + 195

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Using it's concept, it is found that the domain and range of the function are given by:

D: [–4, ∞) and R: [0, ∞)

In this problem, the function is:

[tex]g(x) = \sqrt{x + 4}[/tex]

The square root function does not exist for negative values, hence the domain is represented by:

[tex]x + 4 \geq 0 \rightarrow x \geq -4[/tex]

The range of the square root function is [tex]x \geq 0[/tex], which remains the same as there are no vertical translations.

Hence the correct option is given by:

D: [–4, ∞) and R: [0, ∞)

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Answer: C) D: [–4, ∞) and R: [0, ∞)

Step-by-step explanation:

The amount of money that you will need to pay them back after the 6 months is $2,070.

Given the following data:

Interest rate = 7% = 0.07.

Simple interest = $2000.

Time = 6 months.

Conversion:

12 months = 1 year.

6 months = 0.5 year.

Mathematically, simple interest is given by this formula:

[tex]I=PRT[/tex]

Where:

Substituting the given parameters into the formula, we have;

[tex]I=2000 \times 0.07 \times 0.5[/tex]

I = $70.

Amount to be paid;

[tex]A=I+P\\\\\A=70+2000[/tex]

A = $2,070.

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Answer:

exuse me but i dont see the questions

Step-by-step explanation:

Answer:

a

Step-by-step explanation:

the solution to the system is at the point of intersection of the linear equations.

the lines intersect at (- 4, 3 )

then solution is (- 4, 3 )

Answer:

Add 509 and 476; The Roman Era lasted for a total of 985 years from 509 B.C to 476 A.D.

Step-by-step explanation:

Answer:

[tex]A_{Circle} \approx 39.24\: units^2[/tex]

Step-by-step explanation:

Square ABCD is inscribed in a circle with center P such that BC = 5 units.

BD will be diagonal of the square as well as diameter of the circle.

[tex]BD = BC\times\sqrt 2[/tex]

[tex]\implies BD = 5\sqrt 2[/tex]

-> Diameter of the circle [tex](d) = 5\sqrt 2[/tex]

-> Radius of the circle [tex](r) = 2.5\sqrt 2=3.535\: units[/tex]

[tex]A_{Circle} =\pi(r)^2[/tex]

[tex]\implies A_{Circle} =3.14(3.535)^2[/tex]

[tex]\implies A_{Circle} =3.14(3.535)^2[/tex]

[tex]\implies A_{Circle} =39.2381465[/tex]

[tex]\implies A_{Circle} \approx 39.24\: units^2[/tex]

The probability helps us to know the chances of an event occurring. The probability of event n occurring is 0.3.

The probability helps us to know the chances of an event occurring.

[tex]\rm{Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}[/tex]

We know that for two independent events the probability of the two events occurring simultaneously is written as,

[tex]P(A \cap B) = P(A)\cdot P(B)[/tex]

Now, as we know the probability of event m occurring also we know the probability of both events occurring simultaneously, therefore, the probability of event n occurring can be written as,

[tex]P(A \cap B) = P(A)\cdot P(B)\\\\P(m \cap n) = P(m)\cdot P(n)\\\\0.138 = 0.46 \times P(n)\\\\P(n) = 0.3[/tex]

Hence, the probability of event n occurring is 0.3.

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Answer:

I think the answers is 27, just add all of those squares.

Answer:

[tex]\pi (\frac{diameter}{2} )^{2}[/tex]

Step-by-step explanation:

The formula is pretty similar to the standard equation, π·r², since the diameter is just the radius times two.

Hope that helps! :)

Answer: See below

Step-by-step explanation:

a) There is a correlation between the number of employees in the plant and the number of products produced yearly. Specifically, a positive correlation exists because, as we can see on the table, as the number of employees increases, the number of products also increases. And the rate of increase is constant.

b) Let the function be: y = mx + b

When x = 0; y = 120

So:

120 = 0 + c

c = 120

Now the slope:

[tex]\text { Slope }=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{200-120}{10-0}=\frac{80}{10}=8[/tex]

Therefore, the equation that best fits the data is y = 8x + 120

c) The slope in the function represents the constant rate of change, meaning that as the number of employees increases by 1, the number of products produced monthly increases by 20. While the y-intercept of the plot, which is 120, indicates the constant number of products, that is to say, when there are no employees, there are still 120 products.