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 10,800ð ft3.
h=
let's assume that the silo will be totally filled including the hemisphere.
radius of cylinder and hemisphere is 15
let the height up to the hemisphere be h ft
πr^2 h + (1/2) (4/3)πr^3 = volume
π(15)^2 h + (2/3)π (15)^3 = 10800
225πh + 2250π = 10800
h = (10800 - 2250π)/(225π)
= appr 5.28 ft
very strange looking silo
I suspect there is a typo in 10,800ð ft3
Make the necessary correction and repeat the above steps
To find the height of the silo, we need to set up an equation using the volume of the silo. The volume of the silo can be calculated by adding the volume of the cylinder and the volume of the hemisphere.
1. Volume of the cylinder:
The volume of a cylinder can be calculated using the formula V = πr^2h, where V is the volume, r is the radius, and h is the height. Since the diameter of the silo is given as 30 feet, the radius would be half of the diameter, which is 15 feet. So, the volume of the cylinder is Vc = π(15)^2h.
2. Volume of the hemisphere:
The volume of a hemisphere can be calculated using the formula V = (2/3)πr^3, where V is the volume and r is the radius. Using the same radius of 15 feet, the volume of the hemisphere is Vh = (2/3)π(15)^3.
3. Total volume of the silo:
Adding the volume of the cylinder and the volume of the hemisphere, we get the total volume of the silo as Vt = π(15)^2h + (2/3)π(15)^3.
Given that the total volume should be 10,800π ft^3, we can set up the equation:
10,800π = π(15)^2h + (2/3)π(15)^3.
Simplifying the equation:
10,800π = 225πh + 450π(15)^2.
Canceling out π on both sides:
10,800 = 225h + 450(15)^2.
Calculating 450(15)^2:
10,800 = 225h + 450(225).
Simplifying further:
10,800 = 225h + 101,250.
Rearranging the equation:
225h = 10,800 - 101,250.
Calculating 10,800 - 101,250:
225h = -90,450.
Dividing both sides by 225:
h = -90,450 / 225.
Calculating -90,450 / 225:
h = -402.
Therefore, the height of the silo that will result in a capacity of 10,800π ft^3 is 402 feet.
To find the height (h) of the silo, we will first calculate the volume of the silo.
The volume V of a cylinder is given by the formula:
V = π * r^2 * h
where r is the radius and h is the height.
The volume V of a hemisphere is given by the formula:
V = (2/3) * π * r^3
Since the diameter of the silo is 30 feet, the radius r is equal to half of the diameter, which is 30/2 = 15 feet.
Let's calculate the volume of the cylinder part:
V_cylinder = π * r^2 * h
And the volume of the hemisphere part:
V_hemisphere = (2/3) * π * r^3
The total volume of the silo is the sum of the volume of the cylinder and the volume of the hemisphere:
V_total = V_cylinder + V_hemisphere
We need to find the height (h) that will result in a total volume of 10,800π ft^3, so we have the equation:
10,800π = V_total
Substituting the formulas for the volumes, we have:
10,800π = π * r^2 * h + (2/3) * π * r^3
Simplifying:
10,800 = r^2 * h + (2/3) * r^3
Substituting the value of r:
10,800 = (15^2) * h + (2/3) * (15^3)
Simplifying further:
10,800 = 225 * h + 2 * 3375
10,800 = 225h + 6750
Rearranging the equation to solve for h:
225h = 10,800 - 6750
225h = 4050
h = 4050 / 225
h = 18
Therefore, the height (h) of the silo that will result in a capacity of 10,800π ft^3 is 18 feet.