The volume V (r) of a sphere is a function of its radius r. Suppose a spherical snowball with a
radius 2 f t started to melt so that the radius is changing at a constant rate of 4.5 inches per minute.
If f(t) feet is the radius of the snowball after t minutes, do the following:
a. Compute (V ◦ f)(t) and interpret your result.
[Hint: Find V (r) first.]
b. Find the volume of the snowball after 3 minutes
We all know that
V(r) = 4pi/3 r^3
clearly, f(t) = 21-4.5t
V(f(t)) = 4pi/3 (21-4.5t)^3
Now just plug in your numbers
the answer in letter b please
a. To compute (V ◦ f)(t), we need to find the volume function V(r) first.
The volume of a sphere is given by the formula V(r) = (4/3)πr^3, where π is a constant.
Substituting f(t) for r, we have V(f(t)) = (4/3)π(f(t))^3.
Interpreting the result, (V ◦ f)(t) represents the volume of the snowball with radius f(t) at time t.
b. We are asked to find the volume of the snowball after 3 minutes.
Given that the radius is changing at a constant rate of 4.5 inches per minute, we can express f(t) as:
f(t) = 2ft + (4.5 in/min) * t.
After 3 minutes, the radius would be:
f(3) = 2ft + (4.5 in/min) * 3.
Simplifying the expression, we have:
f(3) = 2ft + 13.5 in.
Using this value of f(3), we can substitute it into the volume function V(r) to find the volume of the snowball after 3 minutes:
V(f(3)) = (4/3)π(f(3))^3.
To answer these questions, we need to follow the given information and use the formulas for the volume of a sphere.
a. Compute (V ◦ f)(t) and interpret your result:
First, let's find the formula for the volume of a sphere. The volume of a sphere with radius r is given by the formula: V(r) = (4/3) * π * r^3.
Now, since the radius of the spherical snowball is changing at a rate of 4.5 inches per minute, we can say that f(t) = 2 + (4.5 * t/60), where t is the time in minutes.
To find (V ◦ f)(t), we need to substitute f(t) into the formula for V(r):
(V ◦ f)(t) = V(f(t)) = (4/3) * π * (f(t))^3
Substituting f(t) = 2 + (4.5 * t/60), we get:
(V ◦ f)(t) = (4/3) * π * (2 + (4.5 * t/60))^3
This expression represents the volume of the snowball as a function of time.
b. Find the volume of the snowball after 3 minutes:
To find the volume of the snowball after 3 minutes, we can substitute t = 3 into the expression we obtained in part a.
(V ◦ f)(3) = (4/3) * π * (2 + (4.5 * 3/60))^3
Now we can calculate this expression to find the volume of the snowball after 3 minutes.