Question

a marble dropped in still water will create circular ripples or waves. the radius of each circular wave will increase at 4 centimeters per second. what is the circumference of the circle after t seconds? options: c(t) = πt², c(t) = 2πt, c(t) = 8πt, c(t) = 16πt²

Explanation:

Step1: Determine the radius after t seconds

The radius increases at 4 centimeters per second, so after \( t \) seconds, the radius \( r(t) = 4t \) (since initial radius is 0 when \( t = 0 \), and it grows linearly with rate 4 cm/s).

Step2: Recall the formula for circumference of a circle

The formula for the circumference \( C \) of a circle is \( C = 2\pi r \), where \( r \) is the radius.

Step3: Substitute the radius function into the circumference formula

Substitute \( r(t) = 4t \) into \( C = 2\pi r \). So \( C(t)=2\pi(4t)=8\pi t \).

Answer:

\( C(t) = 8\pi t \) (the third option)