Suppose a rock dropped into a pond creating a circular ripple travelling at a speed of $80 \mathrm{~cm} / \mathrm{s}$ in an outward direction.
a) Let $r$ be the radius of the circle. Express $r$ as a function of time, $t$, in seconds.
b) If $A$ is the area of this circle as a function of the radius, find $A \circ r$ and interpret it.
Answer
\(\boxed{\text{Final Answer:}}\) a) The radius of the circle as a function of time, \(t\), in seconds is \(r(t) = 80t\). b) The area of the circle as a function of time is \(A(t) = \pi (80t)^2\). This represents how the area of the ripple increases as time passes after the rock is dropped into the pond.
Step 1 ：Suppose a rock dropped into a pond creating a circular ripple travelling at a speed of \(80 \mathrm{~cm} / \mathrm{s}\) in an outward direction.
Step 2 ：Let \(r\) be the radius of the circle. The radius of the circle is directly proportional to the time since the rock was dropped. Therefore, we can express the radius as a function of time as follows: \(r(t) = 80t\).
Step 3 ：If \(A\) is the area of this circle as a function of the radius, the area of a circle is given by the formula \(A = \pi r^2\).
Step 4 ：Substitute \(r(t) = 80t\) into the formula for the area of a circle, we get \(A(t) = \pi (80t)^2\). This represents the area of the circle as a function of time.
Step 5 ：\(\boxed{\text{Final Answer:}}\) a) The radius of the circle as a function of time, \(t\), in seconds is \(r(t) = 80t\). b) The area of the circle as a function of time is \(A(t) = \pi (80t)^2\). This represents how the area of the ripple increases as time passes after the rock is dropped into the pond.
