For the following exercises, consider this scenario: For...

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.) Which forest's population is growing at a faster rate?

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.)
Which forest's population is growing at a faster rate?

Key Concepts

Interpreting Percentage Growth

The parameter r in the exponential growth function, when expressed as a decimal, corresponds directly to the percentage increase per time period. For example, a multiplier of 1.025 corresponds to a 2.5% growth rate per period. This concept is crucial for comparing different exponential functions to determine which one is increasing at a quicker rate.

Exponential Growth Functions

Exponential growth functions describe processes where a quantity increases over time at a rate proportional to its current value. They are generally expressed in the form f(t) = a(1 + r)^t, where a is the initial value, r is the growth rate, and t represents time. This formulation models many real-world phenomena such as populations, investments, and radioactive decay.

Growth Rate Comparison

The growth rate in an exponential function is determined by the multiplier (1 + r). Comparing the growth rates of two exponential functions involves evaluating these multipliers. A larger multiplier indicates a higher rate of increase, meaning that the quantity represented by that function grows faster over time.

Transcript

00:01 All right, this is a pretty interesting question.

00:03 For every year t, the population of a forest of trees represented by the function a of t is equal to 115 times 1 .025 to the t power.

00:14 And another forest, it's basically the same type of tree, but it's a different function.

00:19 This time it's b of t is equal to 82 times 1 .029 to the t power.

00:25 And we're basically trying to figure out which forest population grows at a faster rate.

00:30 So let me get rid of this.

00:31 This is from the previous question.

00:32 So we have two fourths.

00:35 We have forced a and b.

00:38 And the first one was, i actually do a is equal to.

00:43 Let's do b is equal two.

00:44 I already forgot the numbers.

00:48 115 times 1 .025 to the t power.

01:00 And the other one, oops, 1 .025.

01:05 And the other one was 82, 1 .029.

01:16 And we're trying to figure out which population has a faster growth.

01:21 So this might be a little bit tricky because you might be looking at the overall number that kind of pops out.

01:29 Right, right off the bat, right? you can see this first one right here.

01:31 It's 115 times 1 .025 versus a really smaller number, 82 times almost the same number, right, times 1 .029.

01:40 So you would probably think, oh, look, a looks like it's a bigger number, right? is 115 versus, you know, only 82, so that's probably the one that's increasing at a faster rate.

01:50 However, this is not true for one reason.

01:55 Sorry, i had to take a call really quickly.

01:57 It is paused the recording.

01:58 So anyway, so what i was saying was you might think that a, right, 115 times 1 .02, 1 .025, has a bigger number compared to 82 times 1 .029, because, you know, 115 looks much bigger than 82...

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