The radius, r, in centimeters, of a melting snowball is given by r=55−1.5t, where t is the time in hours. The V=4/3πr^3 cm^3. Find a formula for V=f(t), the volume of the snowball as a function of time.
well, just plug it in:
v = 4/3 πr^3 = 4π/3 (55-1.5t)^3
To find a formula for V=f(t), the volume of the snowball as a function of time, we need to substitute the expression for r into the formula for volume V.
Given:
r = 55 - 1.5t
Formula for volume of a sphere:
V = (4/3)πr^3
Replacing r with 55 - 1.5t, we get:
V = (4/3)π(55 - 1.5t)^3
Simplifying further:
V = (4/3)π(55 - 1.5t)(55 - 1.5t)(55 - 1.5t)
V = (4/3)π(55 - 1.5t)(3025 - 165t + 2.25t^2)
Expanding and multiplying:
V = (4/3)π(3025 - 165t + 2.25t^2 - 82.5t + 4.5t^2 + 2.25t - 1.5t^3)
V = (4/3)π(1.5t^3 - 177t + 3025)
Therefore, the formula for V=f(t) is:
V = (4/3)π(1.5t^3 - 177t + 3025)