"Why below question is considered as Linear and not exponential growth? A ball falls from a height of 2 meters onto a firm surface and jumps after each impact each back to 80% of the height from which it fell. ¨ Set up the function, which indicates the height of the ball after the nth impact reached. How high does the ball jump after the 5th impact? Below question is considered as a linear growth and not exponential growth . I dont understand why its linear growth or decay. y=2∗0.85 For the percentage neither it add to 1 nor minus from 1. I wish to ask why it's not y=2∗.2^5 (I took 80% as decay and did 1−.8=.2)"

Julia Chang

Julia Chang

Answered question

2022-09-23

Why below question is considered as Linear and not exponential growth?
A ball falls from a height of 2 meters onto a firm surface and jumps after each impact each back to 80% of the height from which it fell. ¨ Set up the function, which indicates the height of the ball after the nth impact reached. How high does the ball jump after the 5th impact?
Below question is considered as a linear growth and not exponential growth . I dont understand why its linear growth or decay.
y = 2 0.8 5
For the percentage neither it add to 1 nor minus from 1.
I wish to ask why it's not y = 2 .2 5 (I took 80% as decay and did 1−.8=.2)

Answer & Explanation

Jane Acosta

Jane Acosta

Beginner2022-09-24Added 14 answers

With each impact, the height of the bounce decreases to 80% of what it was. But you've made it decrease by 80% (to 20%) by subtracting the 80% from 1.
I think that's where your confusion is.
You need to multiply the height by 80% with each bounce, and the decay comes from the fact that 80% is smaller than 1.
Now, looking at it step by step:
After no bounces (the starting position), you've multiplied by 0.8 no times, so the height in metres is
y = 2 = 2 0.8 0
( 0.8 0 simply means "Don't multiply by 0.8 at all": anything to the power of 0 is 1. But putting it in helps to show the pattern.)
After 1 bounce, you've multiplied by 0.8 once, so
y = 2 0.8 = 2 0.8 1
After 2 bounces, you've multiplied by 0.8 twice, so now
y = 2 0.8 0.8 = 2 0.8 2
After 3 bounces, you've multiplied by 0.8 3 times, making
y = 2 0.8 0.8 0.8 = 2 0.8 3
After n bounces, you've multiplied by 0.8 n times, so
y = 2 0.8 n
which is the equation the question wants you to use.
Then just put n=5 to get the height after 5 bounces.
deiluefniwf

deiluefniwf

Beginner2022-09-25Added 4 answers

Edit: When a problem like this is described in words, "of" is very often the same as "multiplied by": e.g. "two thirds of x" means 2 3 x and "80% of y" means 80%∗y.

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