Can you give me step by step instruction on how to solve this problem
Helium is pumped into a spherical balloon at a constant rate of 4cubic feet per second. How fast is the radius increasing after 3 minutes? At what time (if any) is the radius increasing at a rate of 120 feet per second?
given: dV/dt = 4 ft^3/sec
V= (4/3)πr^3
dV/dt = 4πr^2 dr/dt
3 min = 180 sec
So after 3 min, V = 180(4) or 720 ft^3
720 = (4/3)πr^3
solve for r, then sub into
dV/dt = 4πr^2 dr/dt
solve for dr/dt
For the second part, replace dr/dt by 120
That way you will be able to solve for r
Using that r, find the Volume,
divide that volume by 4 to get the number of seconds it took to fill it that much.
To solve this problem, we can use the formula for the volume of a sphere, which is V = (4/3)*π*r^3, where V is the volume and r is the radius.
Step 1: Convert the given rate of pumping helium from cubic feet per second to gallons per second. Since 1 cubic foot is equal to 7.48 gallons, the rate of pumping helium is 4 cubic feet per second * 7.48 gallons per cubic foot = 29.92 gallons per second.
Step 2: Convert the given time of 3 minutes to seconds. Since 1 minute is equal to 60 seconds, 3 minutes is equal to 3 minutes * 60 seconds per minute = 180 seconds.
Step 3: Differentiate the formula for the volume of a sphere with respect to time to find the rate at which the volume changes with respect to time, and then differentiate the formula for the volume with respect to the radius to find the rate at which the radius changes with respect to time.
dV/dt = (4/3)*π*(3r^2 * dr/dt), where dV/dt represents the rate of change of volume with respect to time and dr/dt represents the rate of change of radius with respect to time.
Step 4: Substitute the given values and solve for dr/dt.
29.92 gallons per second = (4/3)*π*(3r^2 * dr/dt)
dr/dt = (29.92 gallons per second) / ((4/3)*π*(3r^2))
dr/dt = (29.92/(4/3)) / (π*(3r^2))
dr/dt = (22.44) / (π*(3r^2))
dr/dt = 7.51 / (π*r^2)
Step 5: Substitute the time value of 180 seconds into the expression for dr/dt to find the rate at which the radius is increasing after 3 minutes.
dr/dt = 7.51 / (π*r^2)
dr/dt = 7.51 / (π*(r^2))
dr/dt = 7.51 / (π*(r^2))
dr/dt = 7.51 / (π*(r^2))
dr/dt ≈ 7.51 / (3.14159265359*(r^2))
dr/dt ≈ 0.76 / (r^2)
Step 6: Substitute the desired rate of 120 feet per second into the expression for dr/dt to find the time at which the radius is increasing at that rate.
120 = 0.76 / (r^2)
(r^2) = 0.76 / 120
(r^2) ≈ 0.0063
r ≈ sqrt(0.0063)
r ≈ 0.08
Step 7: Since the radius cannot be negative, we disregard the negative value.
The radius is increasing at a rate of approximately 0.76 feet per second after 3 minutes (180 seconds). The radius is increasing at a rate of 120 feet per second when the radius is approximately 0.08 feet.
Note: It's important to note that the calculations provided are examples only, and you may need to use more accurate values for π to get a more precise result.