A spherical balloon is being inflated so that its volume is increasing
at a rate of 5 cubic feet /min.. At what rate is the diameter increasing
when the diameter is 12 feet?
surface area of sphere = 4pi R^2, R = D/2 = 6
so
dV = 4 pi r^2 dr
dV/dt = 4pi r^2 dr/dt
5 ft^3/min = 4 pi (36) dr/dt
dr/dt = 5/ { 144 pi } = 0.011 ft/min
D = 2 r
so
d D/dt = 2 dR/dt
= 0.022 ft/min
To find the rate at which the diameter is increasing, we first need to relate the volume of the sphere to its diameter. The volume (V) of a sphere with diameter (d) can be expressed as:
V = (4/3)πr^3,
where r is the radius of the sphere.
Since the radius (r) is half the diameter (d), we can express the volume in terms of the diameter as:
V = (4/3)π(d/2)^3.
Differentiating both sides of the equation with respect to time (t), we get:
dV/dt = (4/3)π(3/2)(d/2)^2 * (dd/dt).
Given that dV/dt = 5 cubic feet/min, we need to find dd/dt when d = 12 feet.
Substituting the given values into the equation, we get:
5 = (4/3)π(3/2)(12/2)^2 * (dd/dt).
Simplifying the equation, we have:
5 = 4π(3/2)(6/2)^2 * (dd/dt).
5 = 4π(3/2)(9) * (dd/dt).
5 = 54π * (dd/dt).
To solve for dd/dt, we divide both sides of the equation by 54π:
dd/dt = 5 / (54π).
Therefore, the rate at which the diameter is increasing when the diameter is 12 feet is given by dd/dt = 5 / (54π).
To find the rate at which the diameter is increasing, we need to use the related rates formula.
Let's denote the diameter of the balloon as D and the volume as V. We are given that dV/dt = 5 cubic feet/min and we need to find dD/dt when D = 12 feet.
The formula relating volume and diameter for a sphere is V = (4/3) * π * (D/2)^3.
Taking the derivative with respect to t on both sides of the equation using the chain rule, we get:
dV/dt = (4/3) * π * 3 * (D/2)^2 * (dD/dt)
Now, we can substitute the given values into the equation to find the rate at which the diameter is increasing:
5 = (4/3) * π * 3 * (12/2)^2 * (dD/dt)
Simplifying the equation gives:
5 = 4π * 36 * (dD/dt)
Now we can solve for dD/dt:
dD/dt = 5 / (4π * 36)
Evaluating this expression gives:
dD/dt ≈ 0.011 feet/min
Therefore, when the diameter is 12 feet, it is increasing at a rate of approximately 0.011 feet/min.