**Problem 1 :**

(a) Find the surface area of the sphere shown below.

(b) When the radius doubles, does the surface area double ?

**Problem 2 :**

The circumference of a great circle of a sphere is 13.8π feet. What is the surface area of the sphere ?

**Problem 3 : **

Find the volume of the sphere shown below.

**Problem 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 ?

**Problem 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.

**Problem 1 :**

(a) Find the surface area of the sphere shown below.

(b) When the radius doubles, does the surface area double ?

**Solution :**

**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.

**Problem 2 :**

The circumference of a great circle of a sphere is 13.8π feet. What is the surface area of the sphere ?

**Solution : **

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.

**Problem 3 : **

Find the volume of the sphere shown below.

**Solution :**

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.

**Problem 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 ?

**Solution : **

**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.

**Problem 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.

**Solution :**

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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