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 $$n$$th impact reached. How high does the ball jump after the $$5$$th 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 $$1-.8=.2$$)

Here's what you've done.

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.

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 $$\frac23*x$$ and "$$80\%$$ of $$y$$" means $$80\%*y$$.

• My question is why its linear growth and not an exponential growth – tomtom Dec 16 '18 at 8:56
• It's not linear—that would mean adding or subtracting a fixed amount with each bounce. It's exponential because we're multiplying by a fixed amount ($0.8$) each time, and it's decay because repeatedly multiplying something by $0.8$ makes it get smaller and smaller. If we were multiplying by something bigger than $1$ (e. g. $1.1$ in $y=a*1.1^n$) then it would be growth. – timtfj Dec 16 '18 at 11:40
• i am quite lost . eg the value of a new car in 2015 is 4000. it depreciates 7% every year. how much will the car be worth in 2024. here i notice that solution is first find (1-0.07)= 0.93 and then 4000(0.93)^9 is the answer. may i ask why some time for decay we minus decaly percentage from 1 and in above we just dont . may i ask why – tomtom Dec 16 '18 at 20:03
• If it depreciates by $7\%$, it goes down to $93\%$ of what it was. So, multiply by $0.93$, $9$ times (for the $9$ years): $4000*0.93^9=2081.64$ – timtfj Dec 16 '18 at 20:13
• It's the difference between decreasing by a certain amount (which needs subtracting from $100\%$) and decreasing to a certain amount. The key thing is to work out what multiple the new value is of the old one. – timtfj Dec 16 '18 at 20:17