I posted this earlier but the number was wrong. I don't know why it didn't post the other pie symbol. And ofcourse the extra pie throws me off!

A large grain silo is to be constructed in the shape of a circular cylinder with a hemisphere attached to the top (see the figure). The diameter of the silo is to be 30 feet, but the height is yet to be determined. Find the height h of the silo that will result in a capacity of 10,800(pie) ft3.

h=

PI, not pie!

v = pi r^2 h + 2/3 pi r^3
So, given the numbers above,

pi (225) h + 2/3 pi (3375) = 10800 pi
h = (10800pi - 2250 pi)/(225pi) = 38

Steve,

Sorry for my mistake on the PI, I was in a hurry. But I placed the answer you provided and it's wrong?

To find the height (h) of the silo, we need to use the volume formula for a cylinder and hemisphere combined:

Volume of cylinder + Volume of hemisphere = Total volume of silo

Given:
Diameter of silo = 30 feet
Total volume of silo = 10,800 π ft³

Let's break down the problem step by step:

1. Find the radius (r) of the silo:
- Given the diameter (d) of the silo is 30 feet, the radius (r) can be calculated as half the diameter.
- Therefore, r = d/2 = 30/2 = 15 feet.

2. Calculate the volume of the cylinder:
- The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height.
- We need to use the radius (r) we calculated earlier and the total volume of the silo (10,800 π ft³) to find the volume of the cylinder.
- V_cylinder = π * 15² * h = 225πh ft³.

3. Calculate the volume of the hemisphere:
- The volume of a hemisphere is given by the formula V = (2/3)πr³, where r is the radius.
- Since the hemisphere is attached to the top of the cylinder, its volume needs to be subtracted from the total volume of the silo.
- V_hemisphere = (2/3) * π * 15³ = 4500π/3 ft³.

4. Set up the equation:
- We can now set up the equation by adding the volume of the cylinder and subtracting the volume of the hemisphere from the total volume of the silo.
- 225πh + 4500π/3 = 10,800π.
- Simplifying the equation, we get: 225h + 4500/3 = 10,800.
- Dividing each term by 225, we have: h + 20 = 48.
- Subtracting 20 from each side, we get: h = 48 - 20 = 28 feet.

Therefore, the height (h) of the silo that will result in a capacity of 10,800 π ft³ is 28 feet.