INFORMATION:
We know that:
- A rain drop hitting a lake makes a circular ripple
- the radius, in inches, grows as a function of time in minutes according to r(t) = 25√(t+2)
And we must find the area of the ripple as a function of time and the area of the ripple at t=2
STEP BY STEP EXPLANATION:
To find it, we must:
1. Write the equation for the area of a circle
[tex]A=\pi\times r^2[/tex]
2. Write the given equation for the radius
[tex]r(t)=25\sqrt{t+2}[/tex]
3. Replace the equation for the radius in the equation for the area of a circle
[tex]A(t)=\pi\times(25\sqrt{t+2})^2[/tex]
4. Find the area when t = 2 replacing it in the function for the area
[tex]\begin{gathered} A(2)=\pi\times(25\sqrt{2+2})^2 \\ \text{ Simplifying, } \\ A(2)=\pi\times(25\sqrt{4})^2 \\ A(2)=\pi\times50^2 \\ A(2)=2500\pi \end{gathered}[/tex]
Finally, the area of the ripple at t=2 is 2500Ï€ in^2
ANSWER:
Area of the ripple as a function of time:
[tex]A(t)=\pi\times(25\sqrt{t+2})^2[/tex]
Area of the ripple at t=2:
A = 2500Ï€in ^2
