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]
