A balloon is in the shape of a cylinder with a hemisphere at both ends.The height of the cylinder is 4 times that of the radius.Air is pumped in the balloon at a rate of 60cm cubed per second .At what rate does the surface area change if the radius is 8cm?
the figure is really just a sphere of radius r and a cylinder of radius r and height 4r. So, the volume is
V = 4/3 π r^3 + πr^2 (4r)
= 16/3 π r^3
So,
dV/dt = 16πr^2 dr/dt
That means that
dr dt = dV/dt / (16πr^2)
The surface area is
A = 4πr^2 + 2πr*4r
= 4πr^2 + 8πr^2
= 12πr^2
Thus
dA/dt = 24πr dr/dt
= 24πr * dV/dt / (16πr^2)
= 3/(2r) dV/dt
Now, just plug in r=8 and dV/dt = 60, giving
dA/dt = 3/16 * 60 = 11.25 cm^2/s
To solve this problem, we need to find the rate of change of the surface area with respect to time, given the rate of change of volume with respect to time.
1. Let's first find the formula for the surface area of the balloon. The surface area of the cylinder is given by A_c = 2πrh, where r is the radius and h is the height. The surface area of each hemisphere is given by A_h = 2πr^2. Therefore, the total surface area is A = A_c + 2A_h.
2. We are given that the height of the cylinder is 4 times the radius. If the radius is 8 cm, then the height is 4 * 8 = 32 cm.
3. Now, let's find the values for the surface area and volume when the radius is 8 cm. Substitute the values into the formulas:
A_c = 2πrh = 2π(8)(32) = 512π cm^2
A_h = 2πr^2 = 2π(8)^2 = 256π cm^2
So, the total surface area is A = A_c + 2A_h = 512π + 2(256π) = 1024π cm^2.
4. We are given the rate of change of volume with respect to time, which is 60 cm^3/s.
5. Let's find the formula for the volume of the balloon. The volume of the cylinder is given by V_c = πr^2h, and the volume of each hemisphere is given by V_h = (2/3)πr^3. Therefore, the total volume is V = V_c + 2V_h.
6. Substitute the values into the formulas:
V_c = πr^2h = π(8)^2(32) = 2048π cm^3
V_h = (2/3)πr^3 = (2/3)π(8)^3 = 1024π cm^3
So, the total volume is V = V_c + 2V_h = 2048π + 2(1024π) = 4096π cm^3.
7. To find the rate of change of the surface area, we need to differentiate the formula for the surface area with respect to time. Since both the radius and the height are changing with respect to time, we need to use the chain rule.
dA/dt = dA/dr * dr/dt + dA/dh * dh/dt
8. We are given that the radius is 8 cm, so dr/dt = 0 (the rate of change of the radius is 0).
9. We need to find dA/dh and dh/dt. We know that dh/dr = 4 (the height is 4 times the radius), so dh/dt = 4 * dr/dt = 0.
10. We also know that dV/dt = 60 cm^3/s, so we can use it to find dA/dh:
dV/dt = dV/dr * dr/dh * dh/dt
60 = dV/dr * (1/4) * 0
dV/dr = 0
11. Since dA/dr = 0, dA/dt = dA/dh * dh/dt = (dA/dh)(0) = 0.
Therefore, the rate of change of the surface area is 0 cm^2/s when the radius is 8 cm.
To find the rate at which the surface area is changing, we need to find the derivative of the surface area with respect to time, and then substitute the given values to find the specific rate.
Let's first find the formula for the surface area of the balloon. The surface area of the cylinder is given by the formula:
A_cylinder = 2πr_cylinder * h_cylinder
where r_cylinder is the radius of the cylinder and h_cylinder is the height of the cylinder.
The surface area of each hemisphere is given by the formula:
A_hemisphere = 2πr^2_hemisphere
where r_hemisphere is the radius of the hemisphere.
Since the height of the cylinder is 4 times the radius, we have:
h_cylinder = 4 * r_cylinder
Thus, the surface area of the cylinder is:
A_cylinder = 2πr_cylinder * 4r_cylinder = 8πr_cylinder^2
The total surface area of the balloon is the sum of the surface area of the cylinder and the surface area of the two hemispheres:
A_total = A_cylinder + 2 * A_hemisphere
Substituting the formulas we derived earlier:
A_total = 8πr_cylinder^2 + 2 * 2πr_hemisphere^2
= 8πr_cylinder^2 + 4πr_hemisphere^2
Since we are given the rate at which air is pumped into the balloon, we can express the change in volume with respect to time:
dV/dt = 60 cm^3/s
The volume of the balloon can be expressed as:
V = A_cylinder * h_cylinder + 2 * (1/2 * A_hemisphere * r_hemisphere)
= A_cylinder * h_cylinder + A_hemisphere * r_hemisphere
= 8πr_cylinder^2 * 4r_cylinder + 2 * (1/2 * 2πr_hemisphere^2 * r_hemisphere)
= 32πr_cylinder^3 + 4πr_hemisphere^3
Differentiating both sides of this equation with respect to time:
dV/dt = d(32πr_cylinder^3)/dt + d(4πr_hemisphere^3)/dt
The derivative of the volume with respect to time gives us the rate at which the volume is changing.
Since we are given the rate at which the volume is changing (60 cm^3/s), we can now substitute the values of r_cylinder, r_hemisphere, and dV/dt into the equation to find dA_total/dt.