Charlie is making a toy silo for his children. The silos height is three times it's radius. He is also adding a dome shaped top that is a half sphere. If the total surface area of the silo, when assembled is 100 square inches , what is the height of the silo(i.e., what is the distance from the base of the silo to the tip of the dome shaped top)?
I see the surface area as the
circular area of the base + rectangle + 1/2 surface area of the sphere
let the radius be r and the height be h
so we want to measure h+r
given:
" the silo's height i three times its' radius" ---> h+r = 3r
h = 2r
πr^2 + 2πrh + (1/2) 4πr^2 = 100
πr^2 + 2πr(2r) + 2πr^2 = 100
7πr^2 = 100
r^2 = 100/(7π)
r = 2.1324 inches
h = 4.2648 inches
so the height of the toy silo = r+h = 6.397
or appr 6.4 inches
To solve this problem, let's break it down step by step.
Step 1: Determine the surface area of each part of the silo.
- The surface area of a cylinder is given by the formula: 2πrh, where r is the radius and h is the height.
- The surface area of a half sphere is given by the formula: 2πr², where r is the radius.
Step 2: Set up the equation for the surface area of the entire silo.
- The total surface area of the silo is the sum of the surface area of the cylinder and the half sphere top.
Step 3: Solve the equation for the unknown variable, which is the radius.
Let's calculate the solution step by step:
Step 1: Determine the surface area of each part of the silo
Let's assume the radius of the base of the silo is denoted as 'r' and the height of the silo is denoted as 'h'.
The surface area of the cylinder is given by: 2πrh
The surface area of the half sphere (dome-shaped top) is given by: 2πr²
Step 2: Set up the equation for the surface area of the entire silo
The total surface area of the silo is the sum of the surface area of the cylinder and the half sphere top:
2πrh + 2πr² = 100 (since the total surface area is given as 100 square inches)
Step 3: Solve the equation for the unknown variable (radius or height)
Since the height of the silo is three times the radius (h = 3r), we can substitute h in terms of r in the equation:
2πr(3r) + 2πr² = 100
Simplifying the equation:
6πr² + 2πr² = 100
8πr² = 100
πr² = 100/8
πr² = 12.5
Now, divide both sides of the equation by π to find the value of r²:
r² = 12.5/π
r² ≈ 3.98
r ≈ √3.98
r ≈ 1.99 (approximately)
Since we know that h = 3r, we can calculate the height:
h = 3(1.99)
h ≈ 5.97 (approximately)
Therefore, the height of the silo (from the base to the tip of the dome-shaped top) is approximately 5.97 inches.