hydrogen is being pumped into a spherical balloon at the rate of 250 cubic inches per minute. in a and b, you must provide an equation with their proper numerical factors when showing the relationships inquired
A) write an equation showing the relationship between the rates at which the volume of the balloon is changing while the radius is changing?
B. write an equation showing the relationship between the rates at which the volume of the ablloon is changing while the area is changing?
C. at what rate is the radius increasing when the volume of the balloon is 972pi incles?
Start with
V=4/3 PI r^3 then take the derivative of each side with respect to time
dV/dt= 4/3 PI * 3r^2 dr/dt
Now do the same with the surface area equation.
For the last, what is radius when V is 972? Put that r into the dV/dt equation.
A) To find the relationship between the rates at which the volume of the balloon is changing while the radius is changing, we need to use the formula for the volume of a sphere.
The volume of a sphere can be calculated using the following formula:
V = (4/3) * π * r^3
Where V represents the volume and r represents the radius.
To find the rate at which the volume is changing with respect to the radius, we can take the derivative of the volume equation with respect to time (t).
dV/dt = (dV/dr) * (dr/dt)
Where dV/dt represents the rate at which the volume is changing with respect to time, (dV/dr) represents the rate at which the volume is changing with respect to the radius, and (dr/dt) represents the rate at which the radius is changing with respect to time.
B) To find the relationship between the rates at which the volume of the balloon is changing while the area is changing, we need to use the formula for the volume of a sphere in terms of its radius.
The volume of a sphere can also be calculated using the following formula:
V = (4/3) * π * r^3
To find the rate at which the volume is changing with respect to the area, we can take the derivative of the volume equation with respect to the area (A).
dV/dA = (dV/dr) * (dr/dA)
Where dV/dA represents the rate at which the volume is changing with respect to the area, (dV/dr) represents the rate at which the volume is changing with respect to the radius, and (dr/dA) represents the rate at which the radius is changing with respect to the area.
C) To find the rate at which the radius is increasing when the volume of the balloon is 972π inches, we need to use the formula for the volume of a sphere.
The volume of a sphere can be calculated using the following formula:
V = (4/3) * π * r^3
We are given that V = 972π, so we can substitute this value into the equation.
972π = (4/3) * π * r^3
Now, we can solve for the radius (r):
r^3 = (972π * 3) / (4π)
r^3 = 729
Taking the cube root of both sides:
r = 9
Therefore, when the volume of the balloon is 972π inches, the radius is increasing at a rate of 9 inches per unit of time.