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?
- nyphdinmdLv 72 months agoFavorite Answer
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
- 2 months ago
Area of a circle = πr²
Now plug the 30t into the r² value below
A(t) = π(30t)²
A(t) = π(900t²)
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
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
The radius is 210cm.