This one is kinda hard. "A ballon in the shape of a cylinder with hemispherical ends of the same radius as that of the cylinder is shown. The ballon is being inflated at the rate of 261pi cubic centimeters per minute. At the instant the radius of the cylinder is 3 centimeters, the volume of the ballon is 144pi cubic centimeters and the radius of the cylinder is increasing at the rate of 2 centimeters per minute. At this instant, how fast is the height of the entire ballon increasing?"

Question

This one is kinda hard. "A ballon in the shape of a cylinder with hemispherical ends of the same radius as that of the cylinder is shown. The ballon is being inflated at the rate of 261pi cubic centimeters per minute. At the instant the radius of the cylinder is 3 centimeters, the volume of the ballon is 144pi cubic centimeters and the radius of the cylinder is increasing at the rate of 2 centimeters per minute. At this instant, how fast is the height of the entire ballon increasing?"

Answer 1

Thanks that kinda but not really helps me.

Answer 2

To find how fast the height of the entire balloon is increasing, we need to relate the volume of the balloon, the radius of the cylinder, and its height using calculus.

Let's break down the problem step by step:

1. Set up the equation for the volume of the balloon:
The volume of the balloon is a combination of the cylindrical part and two hemispherical ends. The formula for the volume of a cylinder is V = πr^2h, and the formula for the volume of a hemisphere is V = (2/3)πr^3. Since we have two identical hemispheres, the total volume is V = (2/3)πr^3 + πr^2h.

2. Differentiate the volume equation with respect to time (t):
We want to find out how fast the volume is changing with respect to time, so we differentiate both sides of the equation with respect to t.

3. Substitute the given values into the equation:
At the instant when the radius of the cylinder is 3 cm, the volume is 144π cm^3, and the radius is increasing at a rate of 2 cm/min.

4. Solve for dh/dt:
After differentiating the equation and substituting the given values, solve for the rate of change of the height (dh/dt). This will give us the answer to the question.

By following these steps, we can find the rate at which the height of the entire balloon is increasing.

Answer 3

Write the volume equation as a function of radius..

v= PI*r*heightcyclinder + 4/3 PI r^3

Put numbers in...I don't see height in the problem given.

dV/dT= 261

When r=3, V= 144, and dr/dt= 2

Find dh/dt
dv/dt= PI*dr/dt*height + dh/dt*PI*r + 4PIr^2 dr/dt

you can find dh/dt here, once you solve for height in the volume equation