Please help! 50 points!!! and brainliest


Select the correct answer.

What is the nth term of the geometric sequence 4, 8, 16, 32, ...?

A. `a_(n)=2(4)^(n-1)`

B. `a_(n)=4(2)^(n-1)`

C. `a_(n)=2(n)^4`

D. `a_n=4(n)^2`

Answer:

Part a) 0.20 revolutions per foot of distance traveled

Part b) The slope of the graph is [tex]m=5\frac{ft}{rev} [/tex]

Step-by-step explanation:

step 1

Find the slope

we know that

A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form [tex]y/x=k[/tex] or [tex]y=kx[/tex]

In a proportional relationship the constant of proportionality k is equal to the slope m of the line and the line passes through the origin

Let

x ----> the number of revolutions

y ----> the distance traveled in feet

we have the point (2,10) ----> see the graph

Find the value of the constant of proportionality k

[tex]k=y/x=10/2=5\frac{ft}{rev} [/tex]

Remember that in a direct variation the constant k is equal to the slope m

therefore

The slope m is equal to

[tex]m=5\frac{ft}{rev} [/tex]

The linear equation is

[tex]y=5x[/tex]

step 2

How many revolutions does Charmaine make per foot of distance traveled?

For y=1

substitute in the equation and solve for x

[tex]1=5x[/tex]

[tex]x=1/5=0.20\ rev[/tex]

Find the difference:

1400 - 1250 = 150

Divide the difference by the starting value:

150 / 1250 = 0.12

Multiply by 100:

0.12 x 100 = 12% increase.

Answer:

12% Increase.

Step-by-step explanation:

Answer:

8 hours

Step-by-step explanation

The first thing one mus realize here is that this is an inverse proportion question. When one quantity increases, the other decreases. That is the time needed to paint the fence decreases as the number of people who work on it increase. This means therefore that we can set up a proportionality equation:

[tex]t \alpha \frac{k}{n}[/tex]

where t is time and n is the number of people. Given the information in the question we can find the proportionality constant k

[tex]t=\frac{k}{n}[/tex]

[tex]12=\frac{k}{2}[/tex]

so k=24

So how many hours does it take for 3 people

[tex]t=\frac{24}{3} =8[/tex]

Answer:

The percent of the students between 14 and 18 years old is 95% ⇒ answer C

Step-by-step explanation:

* Lets revise the empirical rule

- The Empirical Rule states that almost all data lies within 3

 standard deviations of the mean for a normal distribution.

- 68% of the data falls within one standard deviation.

- 95% of the data lies within two standard deviations.

- 99.7% of the data lies Within three standard deviations

- The empirical rule shows that

# 68% falls within the first standard deviation (µ ± σ)

# 95% within the first two standard deviations (µ ± 2σ)

# 99.7% within the first three standard deviations (µ ± 3σ).

* Lets solve the problem

- The ages of students in a school are normally distributed with

 a mean of 16 years

∴ μ = 16

- The standard deviation is 1 year

∴ σ = 1

- One standard deviation (µ ± σ):

∵ (16 - 1) = 15

∵ (16 + 1) = 17

- Two standard deviations (µ ± 2σ):

∵ (16 - 2×1) = (16 - 2) = 14

∵ (16 + 2×1) = (16 + 2) = 18

- Three standard deviations (µ ± 3σ):

∵ (16 - 3×1) = (16 - 3) = 13

∵ (16 + 3×1) = (16 + 3) = 19

- We need to find the percent of the students between 14 and 18

  years old

∴ The empirical rule shows that 95% of the distribution lies

   within two standard deviation in this case, from 14 to 18

   years old

* The percent of the students between 14 and 18 years old

  is 95%

Answer:

Answer Choice C is correct- 95%

[tex]\cos x=0\\\\x=\dfrac{\pi}{2}+k\pi, k\in\mathbb{Z}[/tex]

The x-values where cos(x) = 0 occur at x = π/2 + nπ, where n is any integer.

To determine the x-values where cos(x) equals 0, we refer to the unit circle, where the cosine of an angle represents the x-coordinate of the corresponding point on the circle.

The cosine function equals 0 at angles where the x-coordinate is 0, which occurs at specific points on the unit circle:

Additionally, cosine is a periodic function with a period of 2π, meaning these values will repeat every 2π. Thus, the general solution for x where cos(x) = 0 is:

x = π/2 + nπ, where n is any integer.

For example: when n=0, x = π/2; when n=1, x = 3π/2; and so on.

Hence, the x-values where cos(x) = 0 are expressed as x = π/2 + nπ for any integer n.

Answer:

Step-by-step explanation:

a) We have been given the points (9,-2) and (9,7)

Use the slope formula to solve this.

m= y2-y1/x2-x1

where (x1,y1)=(9,-2)

and (x2,y2)=(9,7)

m=7-(-2)/9-9

m=7+2/0

m=9/0

Having a zero in the denominator means the slope is undefined. Whenever you have a vertical line your slope is undefined....

b) We have been given (-4,-6) and (5,-6)  

Use the slope formula to solve this.

m= y2-y1/x2-x1

where (x1,y1)=(-4,-6)

and (x2,y2)=(5,-6)

m=-6-(-6)/5-(-6)

m=-6+6/5+6

m=0/11

If the numerator of the fraction is 0, the slope is 0. This will happen if the y value of both points is the same. The graph would be a horizontal line and would indicate that the y value stays constant for every value of x....

Answer:

Part a) 0.63

Part b) 0.22

Part c) 0.68

Step-by-step explanation:

The individual probabilities are calculated as

1) Probability of scoring in first attempt = 70%

2) Probability of missing in first attempt = 30%

3)  Probability of scoring in second attempt provided she scores in first attempt = 90%

4)  Probability  of missing in second attempt provided she scores in first attempt = 10%

5) Probability of scoring in 2nd attempt provided she misses in ist attempt = 50%

6)  Probability of missing in 2nd attempt provided she misses in ist attempt = 50%

Part a)

probability of making the throw exactly  both the times = [tex]P(ist)\times P (2nd)[/tex]

Applying values we get

probability of making the throw exactly  both the times= [tex]0.7\times 0.9=0.63[/tex]

Part b)

She will make it exactly once if

1) She scores in first attempt and misses second

2) She misses in first attempt and scores in  the second attempt

Probability of case 1 = [tex]0.7\times 0.1=0.07[/tex]

Probability of case 2 =[tex] 0.3 \times 0.5 =0.15[/tex]

Thus probability She will make it exactly once equals [tex] 0.15+0.07=0.22[/tex]

Part c)

It is a case of conditional probability

Now by Bayes theorem we have

[tex]P(B|A)=\frac{P(B\cap A)}{P(A)}[/tex]

P(B|A) is the probability of scoring in second attempt provided that she has scored only once

P(B) is the probability of scoring exactly once

Given that she makes it exactly once [tex]\therefore P(A)=0.22[/tex]

[tex]P(B\cap A)=0.15[/tex]

using these values we have

[tex]P(B|A)=\frac{0.15}{0.22}\\\\P(B|A)=0.68[/tex]

Answer:

  339π in³

Step-by-step explanation:

The amount of water is the difference between the volume of the cylinder and the volume of the ball. The appropriate volume formulas are ...

  cylinder V = πr²h

  sphere V = (4/3)πr³

For the given numbers, the volumes are ...

  cylinder V = π(5 in)²(15 in) = 375π in³

  sphere V = (4/3)π(3 in)³ = 36π in³

The water volume is the difference of these ...

  water volume = cylinder V - sphere V

     = 375π in³ - 36π in³ = 339π in³

Answer:

  $557.51

Step-by-step explanation:

A financial calculator tells you the payments are ...

  on $80,000 at 4.75%: $417.32

  on $20,000 at 7.525%: $140.19

Then the total monthly payment is ...

  $417.32 +140.19 = $557.51

_____

You can use the amortization formula to find the payment (A) on principal P at interest rate r for t years to be ...

  A = P(r/12)/(1 -(1+r/12)^(-12t))

I find it takes fewer keystrokes to enter the numbers into a financial calculator. Both give the same result.

Final answer:

Troy's total mortgage payment can be found by calculating the monthly payment for the first mortgage at 80% of the home price with an interest rate of 4.75% and the second mortgage at 20% of the home price with a 7.525% interest rate, then combining these two payments. However, exact figures require the use of an amortization formula or an online mortgage calculator.

Explanation:

To calculate Troy's total mortgage payment for an 80/20 mortgage on a $100,000 house, we need to separate the calculations for the first mortgage (80%) and the second mortgage (20%) because they have different interest rates.

First Mortgage Calculation:

The first mortgage is 80% of the home price, which amounts to $80,000. With an interest rate of 4.75%, his monthly payment can be calculated using the formula for a fixed-rate mortgage.

Second Mortgage Calculation:

The second mortgage is 20% of the home price, equal to $20,000. With a higher interest rate of 7.525%, we again use the formula for a fixed-rate mortgage to find the monthly payment.

Without the actual formula or financial calculator, we cannot compute the exact monthly payments here. Typically, you would use the amortization formula or an online mortgage calculator to find the monthly payments for each part of the mortgage, and then sum them up for the total monthly payment.

Combining the Payments:

Once the monthly payments for both mortgages are calculated, they are added together to determine Troy's total monthly mortgage payment.

The triangles are both similar and congruent triangles

The side lengths of the triangles are given as:

Triangle A = x units, y units and 2 units.

Triangle B = 2 units, x units and y units.

In the above parameters, we can see that the triangles have equal side lengths.

This means that they are congruent by the SSS theorem.

Congruent triangles are always similar.

Hence, the triangles are both similar and congruent triangles

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Answer:

Since there are three pairs of congruent sides, we know the triangles are congruent by the SSS congruence theorem. The corresponding sides of the triangle are also in proportion, so they are also similar by the SSS similarity theorem.

Step-by-step explanation:

Edge 23 sample response

Answer:

n/m = 1.5 is direct variation

v = 1/3 bh is joint variation

y = -7 is constant

vw = -18 is inverse variation

Step-by-step explanation:

* Lets explain the types of variation

- Direct variation is a relationship between two variables that can be

 expressed by an equation in which one variable is equal to a constant

 times the other

# Ex: the equation of a direct variation y = kx , where k is a constant

- Inverse variation is a relationship between two variables in which their

 product is a constant.

- When one variable increases the other decreases in proportion so

 that their product is unchanged

# Ex: the equation of a inverse variation y = k/x , where k is a constant

- Joint variation is a variation where a quantity varies directly as the

 product of two or more other quantities

# Ex: If z varies jointly with respect to x and y, the equation will be of

  the form z = kxy , where k is a constant

- The constant function is a function whose output value (y) is the

  same for every input value (x)

* Lets solve the problem

∵ n/m = 1.5 ⇒ multiply both sides by m

∴ n = 1.5 m

∵ The equation in the shape of y = kx

∴ n/m = 1.5 is direct variation

∵ v = 1/3 bh

∵ 1/3 is a constant

∴ v is varies directly with b and h

∴ v = 1/3 bh is joint variation

∵ y = -7

∴ All output will be -7 for any input values

∴ y = -7 is constant

∵ vw = -18

∵ -18 is constant

∵ The product of the two variables is constant

∴ vw = -18 is inverse variation

Answer:

C. 24.1 ft

Step-by-step explanation:

The side of the wall making the tilt of 75° will represent the hypotenuse

The height of the wall will be represented by the side opposite to the angle 75°

Apply the relationship for sine of angle Ф

Formula to use is ;

Sin Ф=  length of opposite side÷hypotenuse

SinФ=O/H

Sin 75°=O/25

0.96592582628=O/25

O=0.96592582628×25 =24.1 ft

Option 24.1 ft

1. Identify the given information:

   - Length of the wall (hypotenuse, [tex]\( c \)[/tex]) = 25 feet
   - Angle [tex](\( \theta \))[/tex] with the ground = 75°

2. Recall the sine function:
   - Sine of an angle [tex](\( \sin \theta \))[/tex] is the ratio of the length of the opposite side to the hypotenuse.
   - Mathematically, [tex]\( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex].

3. Set up the equation using the sine function:
   [tex]\[ \sin(75^\circ) = \frac{\text{height}}{25} \][/tex]

4. Solve for the height:
   - Multiply both sides by 25 to isolate the height.
   [tex]\[ \text{height} = 25 \times \sin(75^\circ) \][/tex]

5. Calculate [tex]\( \sin(75^\circ) \)[/tex]:
   - This value is approximately 0.9659.

6. Substitute [tex]\( \sin(75^\circ) \)[/tex] into the equation:
   [tex]\[ \text{height} = 25 \times 0.9659 \][/tex]
   [tex]\[ \text{height} \approx 24.148 \][/tex]

7. Determine the closest option:
   - The options given are 25.9 ft, 6.5 ft, 24.1 ft, 93.3 ft.
   - The calculated height is approximately 24.148 feet.
   - The closest option is 24.1 ft.

The leading coefficient is a negative value (-3) so the graph goes down on the right side.

This makes either Graph B or C correct.

Now because the degree ( exponent ) (the ^4) is even both ends of the graph go in the same direction.

This makes Graph B the correct answer.

Answer:

The leading coefficient is a negative value (-3) so the graph goes down on the right side.

This makes either Graph B or C correct.

Now because the degree ( exponent ) (the ^4) is even both ends of the graph go in the same direction.

This makes Graph B the correct answer.

Answer:

-81

Step-by-step explanation:

b=a+3

Subtract 3 from each side

b=a+3

b-3 = a+3-3

b-3 =a

Then subtract b from each side

b-b-3 = a-b

-3 = a-b

We want (a-b) ^4

We know a-b = -3

(-3) ^4

-81

Answer: A and C

Step-by-step explanation: Took the test |

                                                                    \/

The true statement is (c) Initially there were 100 wild flowers growing in the meadow.

The function for the number of wild flowers is given as:

[tex]f(x)=\frac{1000}{1+9e^{-0.4x}}[/tex]

Set x to 0

[tex]f(0)=\frac{1000}{1+9e^{-0.4 * 0}}[/tex]

Evaluate the product

[tex]f(0)=\frac{1000}{1+9e^{0}}[/tex]

Evaluate the exponent

[tex]f(0)=\frac{1000}{1+9}[/tex]

Evaluate the sum

[tex]f(0)=\frac{1000}{10}[/tex]

Evaluate the quotient

[tex]f(0)=100[/tex]

The above represents the initial number of wild flowers

Hence, the true statement is (c) Initially there were 100 wild flowers growing in the meadow.

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Answer:

1km=1000m 160m=0.160x4=0.64

Step-by-step explanation:

Ifl = 160m, then:

P = 4 * 160m\\P = 640m

Thus, the perimeter of the base of the pyramid is 640m.

On the other hand, by definition: 1Km = 1000m

By making a rule of three we have:

1Km ---------> 1000m

x --------------> 640m

Where "x" represents the perimeter of the base of the pyramid in Km.

x = \frac {640 * 1} {1000}\\x = 0.64km

For this case we must convert from meters to kilometers. By definition we have to:

[tex]1km = 1000m[/tex]

The perimeter of the base of the pyramid will be given by the sum of the sides:

[tex]P = 160m + 160m + 160m + 160m = 640m[/tex]

We make a rule of three:

1km ----------> 1000m

x ---------------> 640m

Where "x" represents the equivalent amount in km.

[tex]x = \frac {640} {1000}\\x = 0.64km[/tex]

Answer:

0.64km

Answer:

I see this

"Which relation is a function?

A {(-3,4),(-3,8),(6,8)}

B {(6,4),(-3,8),(6,8)}

C {(-3,4),(3,-8),(3,8)}

D {(-3,4),(3,5),(-3,8)}"

So the answer is none of these.

Please make sure you have the correct problem.

Step-by-step explanation:

A set of points is a function if you have all your x's are different.  That is, all the x's must be distinct. There can be no value of x that appears more than once.

If you look at choice A, this is not a function because the first two points share the same x, which is -3.

Choice B is not a function because the first and last point share the same x, which is 6.

Choice C is not a function because the last two points share the same x, which is 3.

Choice D is not a function because the first and last choice share the same x, which is -3.

None of your choices show a function.

If you don't have that choice you might want to verify you written the problem correctly.

This is what I see:

"Which relation is a function?

A {(-3,4),(-3,8),(6,8)}

B {(6,4),(-3,8),(6,8)}

C {(-3,4),(3,-8),(3,8)}

D {(-3,4),(3,5),(-3,8)}"

Answer:

  a. x^4 +2x^2 -6

  b. 2x^5 +x^4 +4x^3 +2x^2 -12x -6

  c. 6x^5 +3x^4 +12x^3 +6x^2 -36x -18

  d. the area of one piece is 1/3 the area of the board

Step-by-step explanation:

a. The length of each piece will be 1/3 the length of the board. Divide each of the coefficients of the length polynomial by 3:

  piece length = (1/3)(board length) = (1/3)(3x^4 +6x^2 -18) = x^4 +2x^2 -6

__

b. The area is the product of length and width.

  piece area = (width)×(piece length) = (2x +1)(x^4 +2x^2 -6)

  piece area = 2x^5 +x^4 +4x^3 +2x^2 -12x -6

__

d. The board area is 3 times the area of one piece. The area of one piece is 1/3 the area of the board.

__

c. board area = 3×(piece area) = 6x^5 +3x^4 +12x^3 +6x^2 -36x -18

Answer:

a. We can say that P > C, where 'P' represents Peter's earnings and 'C' represents Cindy's earnings.

Given that P = 3h and C = 2h, where h =$4.50. We can say also that 3h > 2h.

b. If Cindy wants to earn at least $14 a day working two hours. Then:

2h ≥  $14

To solve the problem, we just need to solve for 'h':

h ≥ $7

Therefore, se should earn more or equal to $14 per hour.

Answer:

Peter works part time for 3 hours every day and Cindy works part time for 2 hours every day.

Part A:

Peter's earning in 3 hours is = [tex]3\times4.50=13.5[/tex] dollars

Cindy's earnings in 2 hours is = [tex]2\times4.50=9[/tex] dollars

We can define the inequality as: [tex]9<13.50[/tex]

Part B:

Let Cindy's earnings be C and number of hours needed be H.

We have to find her per hour income so that C ≥ 14

As Cindy works 2 hours per day, the inequality becomes 2H ≥ 14

So, we have [tex]H\geq 7[/tex]

This means Cindy's per hour income should be at least $7 per hour so that she earns $14 a day.

Answer:

D. {(8, 1), (4, 1), (0, 1), (–15, 1)}

Step-by-step explanation:

A function can not contain two ordered pairs with the same first elements.

Let us look at the options one by one:

A. {(–4, –6), (–3, –2), (1, –2), (1, 0)}

Not a function because (1, –2), (1, 0) have same first element.

B. {(–2, –12), (–2, 0), (–2, 4), (–2, 11)}

Not a function because all the ordered pairs have the same first element.

C. {(0, 1), (0, 2), (1, 2), (1, 3)}

Not a function because (0, 1), (0, 2) have same first element.

D. {(8, 1), (4, 1), (0, 1), (–15, 1)}

This is a function because all the ordered pairs have different first elements i.e. no repetition in first elements of the ordered pairs

Therefore, option D is correct ..

Answer:

The smaller diameter is [tex]3.9\ cm[/tex]

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

The diameter of the circle is equal to BD

Applying the Pythagoras Theorem

[tex]BD^{2} =BE^{2} +DE^{2}[/tex]

substitute the given values

[tex]BD^{2} =3^{2} +2.5^{2}\\ BD^{2} =15.25\\ BD=3.9\ cm[/tex]

Answer:

3.9

Step-by-step explanation:

use Pythagorean Theorem

plus i took the test :)

Answer:

the point estimate of the true proportion of employees is 0.401

Step-by-step explanation:

Given data

plan A = ( 0.241 , 0.561 )

to find out

the point estimate for estimating the true proportion of employees who prefer that plan

solution

we know that given data 98% confidence interval for the proportion of employees who prefer plan A: (0.241, 0.561) so point estimate for estimating the true proportion o employees who prefer that plan is p^E

and we can say here  

lower limit = [tex]p^{-E}[/tex] = 0.241    ..............1

upper limit = [tex]p^{+E}[/tex]  = 0.561    ...............2

now add these two equations 1 and 2

=  [tex]p^{-E}[/tex] + [tex]p^{+E}[/tex]

=  [tex]p^{-E}[/tex] + [tex]p^{+E}[/tex]  = 0.241 + 0.561

2 p = 0.802

2p = 0.806

p = 0.401

So the point estimate of the true proportion of employees is 0.401

The point estimate in this scenario is 0.401, which is the estimated proportion of employees who prefer Plan A. This is calculated by averaging the lower and upper limits of the given 98% confidence interval.

The question is asking for the point estimate which is a single value used as an estimate of the population parameter. In a confidence interval, the point estimate is the mid-point or center of the interval. Here, the confidence interval given for the proportion of employees preferring Plan A is (0.241, 0.561). To find the point estimate, you average the two end values.

The formula for finding the point estimate is: (Lower limit + Upper limit) / 2.

Applying the values from your question: (0.241 + 0.561) / 2 = 0.401.

So, the point estimate, or the estimated true proportion of employees who prefer Plan A, is 0.401.

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Answer:

  4:9 ≠ 6:13

Step-by-step explanation:

The ratios of corresponding segments will be equal if DE || BC. Yana can compare AD:AB versus AE:AC. She will find they're not equal, as expressed by ...

  4 : 9 ≠ 6 : 13

Answer:

4:9 ≠ 6:13

Step-by-step explanation:

Given,

In triangle ABC,

D ∈ AB, E ∈ AC,

Also, AD = 4 unit, DB = 5 unit, AE = 6 unit, EC = 7 units,

Suppose,

DE ║ BC,

[tex]\because \frac{AD}{AB}=\frac{AD}{AD + DB}=\frac{4}{9}[/tex]

[tex]\frac{AE}{AC}=\frac{AE}{AE+EC}=\frac{6}{6+7}=\frac{6}{13}[/tex]

[tex]\implies \frac{AD}{AB}\neq \frac{AE}{AC}[/tex]

Because,

[tex]\frac{4}{9}\neq \frac{6}{13}[/tex]

Which is a contradiction. ( if a line joining two points of two sides of a triangle is parallel to third sides then the resultant triangles have proportional corresponding sides )

Hence, DE is not parallel to segment BC.

He drives 55 miles per hour for x hours. You want to multiply the number of hours by his speed , so you have 55x.

Then you want to subtract that from the total miles he has to drive.

The equation is Y = 385 - 55x

Answer:

B

Step-by-step explanation:

The "value" of a function is the y-value.

Since we want negative y values, we look at "WHERE" the function is "UNDER" the y-axis.

Looking closely, it looks that function (the dip) is from x = -1 to x = 3

Hence, the function is negative at the interval  -1 ≤ x ≤ 3

B is correct.

Answer: the answer would be choice A

Step-by-step explanation: it passes through the origin making it a proportional relationship

Answer:

  22 ft/s

Step-by-step explanation:

15 mph is 1/4 of 60 mph. At 1/4 the speed, the car will travel 1/4 as far in a second.

  1/4×(88 ft/s) = 22 ft/s

Answer:

22 ft/s

Step-by-step explanation:

At 60 mph a car travels  88 feet per second. There are 22 feet per second that a car  travels at 15 mph.

Answer:

  (-5, 6) → (5, -6)

Step-by-step explanation:

Reflection across the origin negates both coordinates.

  (x, y) → (-x, -y)

  (-5, 6) → (5, -6)

Answer:

[tex]\frac{10}{3}[/tex]

Step-by-step explanation:

Let x be the filled bottles of soda Q,

As per statement,

The filled bottles of soda V = 2x,

Given,

Rate of filling of soda Q = 500 liters per sec,

So, the total volume filled by soda Q in a day = 500 × 86400 = 43200000 liters,

( ∵ 1 day = 86400 second ),

Thus, the volume of a bottle of Soda Q = [tex]\frac{\text{Total volume filled by soda Q}}{\text{filled bottles of soda Q}}[/tex]

[tex]=\frac{43200000}{x}[/tex]

Now, rate of filling of soda V = 300 liters per sec,

So, the total volume filled by soda V in a day = 300 × 86400 = 25920000 liters,

Thus, the volume of a bottle of Soda V

[tex]=\frac{25920000}{2x}[/tex]

Thus, the ratio of the volume of a bottle of Soda Q to a bottle of Soda V

[tex]=\frac{\frac{43200000}{x}}{\frac{25920000}{2x}}[/tex]

[tex]=\frac{10}{3}[/tex]

Answer:

Step-by-step explanation:

This is a geometric sequence so the standard formula for a recursive geometric sequence is

[tex]a_{n}=a_{0}*r^{n-1}[/tex]

We know the heights and the number of bounces needed to achieve that height, but in order to write the recursive formula we need r.

The value of r is found by dividing each value of a bounce by the one before it.  In other words, bounce 1 divided by the starting height gives a value of r=240/300 so r = .8

Bounce 2 divided by bounce 1: 192/240 = .8

So r = .8

Therefore, the formula is

[tex]a_{n}=a_{0}(.8)^{n-1)[/tex] where

aₙ is the height of the ball after the nth bounce,

a₀ is the starting height of the ball,

.8 is the rebound percentage, and

n-1 is the number of bounces minus 1

The first problem basically asks us to find n when the starting height is 175 and the bounce height is less than 8.  I used 7.  Here is the formula filled in with our info:

[tex]7=175(.8)^{n-1}[/tex]

and we need to solve for n.  That requires that we take the natural log of both sides.  Here are the steps:

First, divide both sides by 175 to get

[tex].04=(.8)^{n-1}[/tex]

Next, take the natural log of both sides:

[tex]ln(.04)=ln((.8)^{n-1})[/tex]

The power rule of logs says that we can bring the exponent down in front of the log:

[tex]ln(.04)=n-1(ln(.8))[/tex]

Finding the natural logs of those decimals gives us:

[tex]-3.218876=-.223144(n-1)[/tex]

Divide both sides by -.223144 to get your n-1 value:

n - 1 = 14.4251067

That means that, since the ball is not bouncing 14.425 times, it bounces 14 times to achieve a height less than 8.  Let's see how much less than 8 by checking our answer.  To do this, we will solve for aₙ when x = 14:

[tex]a_{n}=175(.8)^{14}[/tex]

This gives us a height at bounce 14 of 7.697 cm, just under 8!

Now for the next part, we want to use a starting value of 250 and .8 as the rebound height.  We want to find a₄, the height of the 4th bounce.

[tex]a_{4}=250(.8)^{4-1}[/tex]

which simplifies to

[tex]a_{4}=250(.8)^3[/tex]

Do the math on that to find the height of the 4th bounce from a starting height of 250 cm is 128 cm

Answer:

First case

   Recursive formula: [tex]h_n = 0.8 \times h_{n-1}[/tex]

   Explicit formula: [tex]h_n = 300 \times 0.8^{n-1}[/tex]

Second case: 15 bounces are needed

Third case: 128 cm

Step-by-step explanation:

Let's call h to he height of the ball

From the table, the rate is computed as follows:

r = 240/300 = 192/240 = 153.6/192 =  122.88/153.6 = 98.3/122.88 = 0.8

Which means this is a geometric sequence (all quotients are equal).

Recursive formula:

[tex]h_0 = 300[/tex]

[tex]h_n = r \times h_{n-1}[/tex]

[tex]h_n = 0.8 \times h_{n-1}[/tex]

where n refers to the number of bounces

Explicit formula:

[tex]h_n = h_0 \times r^{n-1}[/tex]

[tex]h_n = 300 \times 0.8^{n-1}[/tex]

If this ball is dropped from a height of 175 cm, then

[tex]h_n = 175 \times 0.8^{n-1}[/tex]

If the height must be 8 cm or less:

[tex]8 = 175 \times 0.8^{n-1}[/tex]

[tex]8/175 = 0.8^{n-1}[/tex]

[tex]ln(8/175) = (n-1) ln(0.8)[/tex]

[tex]n = 1 + \frac{ln(8/175)}{ln(0.8)}[/tex]

[tex]n = 14.83[/tex]

which means that 15 bounces are needed.

If this ball is dropped from a height of 250 cm, then

[tex]h_n = 250 \times 0.8^{n-1}[/tex]

For the fourth bounce the height will be:

[tex]h_4 = 250 \times 0.8^{4-1}[/tex]

[tex]h_4 = 128[/tex]