# Composite Functions Word problem?

A pebble dropped in a lake creates a circular wave that travels outward at a speed of 30 cm/s.

a.) Use function composition to derive a function, A(t), that expresses the area of the circular wave as a function of time.

b.) What is the area of the circular wave after 3 seconds?

c.) How long does it take for the area enclosed by the circular wave to be 44100π cm^2?

What is the radius of the wave?

Relevance

Let  v = 30 cm/s.  The wave moves out radially so the radius, r, can be written as a function of time, t as

r(t) = v*t  sine the area of a circle A = pi*r^2 then A(t+ = pi*r(t)^2 = pi*v^2*t^2

For b. plug in numbers

For c set A = 44100*pi = pi*v*T  -->  T = 44100/v

• Part A

Area of a circle =  πr²

r=30t     <------cm/s

Now plug the 30t into the r² value below

A(t) = π(30t)²

A(t) = π(900t²)

Part B

3 seconds is 3t therefore we plug 3 into the t value in the area formula that we derived in part A.

A(t) = π(900(3)²)

A(t) = π(8100)cm^2

Part C

44100π = π900t²

Dive both sides by π and it cancels out.

Divide 44100 by 900

44100/900 = t²

Square both sides and we get 7 seconds.

(44100)^1/2 = (900t² )^1/2

210 = 30t