Mathematics, 12.07.2019 13:10 kfull6027

write the explicit formula for the data.
write a recursive rule for the height of the ball on each successive bounce.
if this ball is dropped from a height of 175 cm, how many times does it bounce before it has a bounce height of less than 8 cm? use the same rebound percentage as in the previous problem.
what is the height of the fourth bounce of this ball if it is dropped from a height of 250 cm? use the same rebound percentage as in the previous problem.

write the explicit formula for the data.write a recursive rule f

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Answer from: tbiles99

Step-by-step explanation:

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

$a_{n}=a_{0}*r^{n-1}$

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

$a_{n}=a_{0}(.8)^{n-1)$ 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:

$7=175(.8)^{n-1}$

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

$.04=(.8)^{n-1}$

Next, take the natural log of both sides:

$ln(.04)=ln((.8)^{n-1})$

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

ln(.04)=n-1(ln(.8))

Finding the natural logs of those decimals gives us:

-3.218876=-.223144(n-1)

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:

$a_{n}=175(.8)^{14}$

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.

$a_{4}=250(.8)^{4-1}$

which simplifies to

$a_{4}=250(.8)^3$

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

Answer from: kaitlyn114433

First case

Recursive formula: $h_n = 0.8 \times h_{n-1}$

Explicit formula: $h_n = 300 \times 0.8^{n-1}$

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:

h_0 = 300

$h_n = r \times h_{n-1}$

$h_n = 0.8 \times h_{n-1}$

where n refers to the number of bounces

Explicit formula:

$h_n = h_0 \times r^{n-1}$

$h_n = 300 \times 0.8^{n-1}$

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

$h_n = 175 \times 0.8^{n-1}$

If the height must be 8 cm or less:

$8 = 175 \times 0.8^{n-1}$

$8/175 = 0.8^{n-1}$

ln(8/175) = (n-1) ln(0.8)

$n = 1 + \frac{ln(8/175)}{ln(0.8)}$

n = 14.83

which means that 15 bounces are needed.

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

$h_n = 250 \times 0.8^{n-1}$

For the fourth bounce the height will be:

$h_4 = 250 \times 0.8^{4-1}$

h_4 = 128

Answer from: Quest

step-by-step explanation:

it will be a cylinder. with a diameter of 8 and 4 single pieces high

Answer from: Quest

With a continuous thought process.

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