Math  /  Algebra

QuestionA population of 80,000 toads is expected to shrink at a rate of 9.2%9.2 \% per year.
What will the toad population be in 20 years? 3632 7360 11,609 72,640

Studdy Solution

STEP 1

What is this asking? If we start with 80,000 toads, and they decrease by 9.2% each year, how many toads will be left after 20 years? Watch out! Don't accidentally *increase* the toad population – we're dealing with a *decrease* here!
Also, make sure to apply the decrease over 20 years, not just one!

STEP 2

1. Calculate the population multiplier.
2. Calculate the toad population after 20 years.

STEP 3

We're told the toad population shrinks by 9.2%9.2\% each year.
This means each year, the population is multiplied by (10.092)(1 - 0.092).
Why? Because we start with 100%100\% of the toads, and then lose 9.2%9.2\% of them.
Subtracting gives us 100%9.2%=90.8%100\% - 9.2\% = 90.8\%, which is 0.9080.908 as a decimal.

STEP 4

So, our **population multiplier** is 10.092=0.9081 - 0.092 = \mathbf{0.908}.
This means each year, the number of toads is multiplied by 0.9080.908.

STEP 5

We **start** with 80,00080,000 toads.
After one year, we multiply by our **population multiplier** to get 80,0000.90880,000 \cdot 0.908.
After two years, we multiply by the **population multiplier** *again*: 80,0000.9080.90880,000 \cdot 0.908 \cdot 0.908, which is the same as 80,000(0.908)280,000 \cdot (0.908)^2.
See the pattern?

STEP 6

After 20 years, we'll have multiplied by 0.9080.908 a total of **20 times**.
So, the toad population after 20 years will be 80,000(0.908)2080,000 \cdot (0.908)^{20}.

STEP 7

Let's **calculate** that: 80,000(0.908)2080,0000.1522812,182.480,000 \cdot (0.908)^{20} \approx 80,000 \cdot 0.15228 \approx 12,182.4.

STEP 8

Since we can't have fractions of toads, we'll round to the nearest whole number.
So, after 20 years, there will be approximately 12,182\mathbf{12,182} toads.

STEP 9

The toad population after 20 years will be approximately 12,182.
Looking at the provided options, the closest answer is 11,609.

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