QuestionFind the function $(A \circ r)(t)$ if the radius $r(t)$ of ripples in a pond is known, where $A(r)=\pi r^{2}$ .

Studdy Solution

STEP 1

Assumptions1. The radius $r$ of the outer circle is a function of time $t$ after the pebble strikes the water, given by $r(t)$ . . The area $A$ of the circle is given by the function $A(r)=\pi r^{}$ .
3. We need to find the function $(A \circ r)(t)$ given the function $r(t)$ .

STEP 2

First, we need to understand what $(A \circ r)(t)$ means. In mathematics, $(A \circ r)(t)$ represents the composition of the functions $A$ and $r$ . It means that we first apply the function $r$ to $t$ , and then apply the function $A$ to the result.

STEP 3

So, we can write $(A \circ r)(t)$ as $A(r(t))$ .

STEP 4

Now, we know that the function $A(r)$ is given by $\pi r^{2}$ . So, we can substitute $r(t)$ into $A(r)$ .

STEP 5

Substitute $r(t)$ into $A(r)$ .
$(A \circ r)(t) = A(r(t)) = \pi (r(t))^{2}$ So, the function $(A \circ r)(t)$ is $\pi (r(t))^{2}$ .

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