Chapter Closure
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Exercise 153 Page 704
a The formula for the volume of a cylinder is V = π r^2 h, where r is the radius of the cylinder's base and h is the cylinder's height. The formula for the volume of a hemisphere is V = 23 π r^3, where r is the sphere's radius.
b The formula for the lateral area of a cylinder is A = 2 π r h, where r is the radius of the cylinder's base and h is the cylinder's height. The formula for the surface area of a hemisphere is V = 2 π r^2, where r is the sphere's radius.
a About 35 600 000 cubic feet.
b About 440 000 square feet.
a We want to find the volume of the Smallville's baseball stadium. The top of the stadium is a hemisphere with a radius of 200 feet. Underneath it there is a cylinder with a base with the same radius and the height of 150 feet.
The volume of this stadium is the sum of volumes of the bottom cylinder and the top hemisphere.
Cylinder's Volume
The volume of a cylinder can be found using the following formula.Hemisphere
The volume of a sphere with a radius r equals 43 π r^3. Since we have a hemisphere, its volume is half the volume of a sphere with the same radius. For this reason, we have the following formula for the volume of a hemisphere with a radius r. V = 2/3 π r^3 Let's substitute 200 for r in the above formula and simplify.V = 2/3 π r^3
Substitute
r= 200
V = 2/3 π ( 200)^3
▼
Simplify right-hand side
CalcPow
Calculate power
V = 2/3 π (8 000 000)
Multiply
Multiply
V = 2/3 (8 000 000)π
MoveLeftFacToNum
a*b/c= a* b/c
V = 2 * 8 000 000/3 π
Multiply
Multiply
V = 16 000 000/3 π
Stadium's Volume
The stadium's volume is a sum of volumes of the bottom cylinder and the top hemisphere. Let's find it! 6 000 000 π + 16 000 000/3 π ≈ 35 600 000 The stadium's volume is about 35 600 000 cubic feet.
b We want to find the surface area of the outside of the Smallville's baseball stadium. The top of the stadium is a hemisphere with a radius of 200 feet. Underneath it there is a cylinder with a base with the same radius and the height of 150 feet.
Since we do not include the base of the cylinder, the surface area we want to find is the sum of the lateral surface area of a cylinder and the curved surface area of the hemisphere.
Cylinder's Lateral Area
The cylinder's lateral area can be found using the following formula.Curved Area of a Hemisphere
The surface area of a sphere with a radius r equals 4 π r^2. Since we have a hemisphere, its curved area is half the surface area of a sphere with the same radius. For this reason, we have the following formula for the curved area of a hemisphere with a radius r. V = 2 π r^2 Let's substitute 200 for r in the above formula and simplify. The curved area of the hemisphere is about 80 000 π square feet.Stadium's Outside Area
The stadium's outside area is a sum of volumes of the bottom cylinder and the top hemisphere. Let's find it! 60 000 π + 80 000 π ≈ 440 000 The stadium's outside area is about 440 000 square feet.
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