Question

The Pentagon is planning to build a new, spherical satellite. As is typical in these cases, the specifications keep changing, so that the size of the satellite keeps growing. In fact, the radius of the planned satellite is growing 0.4 feet per week. Its cost will be $1,100 per cubic foot. At the point when the plans call for a satellite 10 feet in radius, how fast is the cost growing

Answer 1

$553300

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:

`V=(4)/(3) \pi r^3\\\\differentiating\ with\ respect\ to\ t:\\\\(dV)/(dt)= (d)/(dt)((4)/(3) \pi r^3)\\\\ (dV)/(dt)= (4)/(3) \pi(d)/(dt)(r^3)\\\\ (dV)/(dt)= (4)/(3) \pi*3r^2(dr)/(dt) \\\\ (dV)/(dt)= 4\pi r^2(dr)/(dt) \\\\Substituting:\\\\ (dV)/(dt)= 4\pi (10 \ feet)^2(0.4\ feet/week)\\\\ (dV)/(dt)=503\ feet^3/week`

Therefore, the cost of increasing volume = 503 feet³/week * $1100 / feet³ = $553300