Answer:
$553300
Step-by-step explanation:
Let the rate of increase of radius with respect to time be dr / dt. Hence:
dr / dt = 0.4 ft/week
The cost of increasing the radius is $1,100 per cubic foot. We can calculate how fast the cost is growing by determining the rate at which the volume increases with time (dV / dt).
The volume (V) of a spherical object is given by:
[tex]V=\frac{4}{3} \pi r^3\\\\differentiating\ with\ respect\ to\ t:\\\\\frac{dV}{dt}= \frac{d}{dt}(\frac{4}{3} \pi r^3)\\\\ \frac{dV}{dt}= \frac{4}{3} \pi\frac{d}{dt}(r^3)\\\\ \frac{dV}{dt}= \frac{4}{3} \pi*3r^2\frac{dr}{dt} \\\\ \frac{dV}{dt}= 4\pi r^2\frac{dr}{dt} \\\\Substituting:\\\\ \frac{dV}{dt}= 4\pi (10 \ feet)^2(0.4\ feet/week)\\\\ \frac{dV}{dt}=503\ feet^3/week[/tex]
Therefore, the cost of increasing volume = 503 feet³/week * $1100 / feet³ = $553300
