Question 1 :
(a) Find the surface area of the sphere shown below.
(b) When the radius doubles, does the surface area double ?
Question 2 :
The circumference of a great circle of a sphere is 13.8π feet. What is the surface area of the sphere ?
Question 3 :
Find the volume of the sphere shown below.
Question 4 :
A baseball and its leather covering are shown. The baseball has a radius of about 1.45 inches.
a. Estimate the amount of leather used to cover the baseball.
b. The surface of a baseball is sewn from two congruent shapes, each of which resembles two joined circles. How does this relate to the formula for the surface area of a sphere ?
Question 5 :
To make a steel ball bearing, a cylindrical slug is heated and pressed into a spherical shape with the same volume. Find the radius of the ball bearing below.
1. Answer :
Solution (a) :
Formula for surface area of a sphere :
S = 4πr^{2}
Substitute 2 for r.
S = 4π (2^{2})
S = 4π (4)
S = 16π
The surface area of the sphere is 16π square inches.
Solution (b) :
When the radius doubles,
r = 2 ⋅ 2
r = 4 inches
Formula for surface area of a sphere :
S = 4πr^{2}
Substitute 4 for r.
S = 4π (4^{2})
S = 4π (16)
S = 64π in^{2}
Because 16π ⋅ 4 = 64π, the surface area of the sphere in part (b) is four times the surface area of the sphere in part (a).
So, when the radius of a sphere doubles, the surface area does not double.
2. Answer :
Draw a sketch.
Begin by finding the radius of the sphere.
Formula for circumference of a circle :
C = 2πr
Substitute 13.8π for C.
13.8π = 2πr
Divide each side by 2π.
6.9 = r
Formula for surface area of a sphere :
S = 4πr^{2}
Substitute 6.9 for r.
S = 4π(6.9)^{2}
Simplify.
S = 4π( 47.61)
Use calculator.
S ≈ 598 ft^{2}
So, the surface area of the sphere is about 598 square feet.
3. Answer :
Formula for volume of a sphere :
V = 4/3 ⋅ πr^{3}
Substitute 22 for r.
V = 4/3 ⋅ π(22^{3})
Simplify.
V = 4/3 ⋅ π(10648)
V = 42592/3 ⋅ π
Use calculator.
V ≈ 44602 cm^{2}
The volume of the sphere is about 44602 cubic cm.
4. Answer :
Solution (a) :
Because the radius r is about 1.45 inches, the surface area is as follows.
Formula for surface area of a sphere :
S = 4πr^{2}
Substitute 1.45 for r.
S = 4π(1.45^{2})
Simplify.
S = 8.41π
Use calculator.
S ≈ 26.4 in^{2}
So, the amount of leather used to cover the baseball is about 26.4 square inches.
Solution (b) :
Because the covering has two pieces, each resembling two joined circles, then the entire covering consists of four circles with radius r.
The area of a circle of radius r is
A = πr^{2}
So, the area of the covering can be approximated by
4πr^{2}
This is the same as the formula for the surface area of a sphere.
5. Answer :
To find the radius of the ball bearing, first we need to find the volume of the slug.
Use the formula for the volume of a cylinder.
V = πr^{2}h
Substitute 1 for r and 2 for h.
V = π(1)^{2}(2)
Simplify.
V = 2π cm^{3}
To find the radius of the ball bearing, use the formula for the volume of a sphere and solve for r.
Formula for volume of sphere :
V = 4/3 ⋅ πr^{3}
Substitute 2π for V.
2π = 4/3 ⋅ πr^{3}
Multiply each side by 3.
6π = 4πr^{3}
Divide each side by 4π.
1.5 = r^{3}
Use a calculator to take the cube root.
1.14 ≈ r
So, the radius of the ball bearing is about 1.14 centimeters.
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Oct 07, 22 12:25 PM
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