**Finding the volume of a sphere in a real world context :**

We can find the volume of a sphere using the formula of volume of a cylinder.

When we model the volume of a sphere, we will be getting the following result.

Volume of cone = 2/3 · Volume of cylinder

Substitute the formula for volume of cylinder.

Volume of the cone = 2/3 · ∏r**²**h

A sphere always has a height which is equal to twice the radius.

So, substitute 2r for h.

Volume of sphere = 2/3 · ∏r²(2r)

Simplify.

Volume of sphere = 4/3 · ∏r³

In this section, we are going to see, how to use the concept "volume of a sphere" in a real world situation.

**Example 1 :**

Soccer balls come in several different sizes. One of the soccer balls has a diameter of 24 centimeters. What is the volume of this soccer ball ? Round your answer to the nearest tenth. Use the approximate of value of ∏, that is 3.14.

**Solution : **

**Step 1 : **

Because soccer ball is in the shape of sphere, we can use the formula of volume of a sphere to find volume of the soccer ball.

Write the formula to find volume of a sphere.

V = 4/3 · ∏r³ ----- (1)

**Step 2 : **

To find the volume, we need the radius of the sphere. But, the diameter is given, that is 24 cm. So, find the radius.

r = diameter / 2

r = 24/2

r = 12

**Step 3 : **

Plug ∏ ≈ 3.14 and r = 12 in (1).

V ≈ 4/3 · 3.14 · 12³

Simplify.

V ≈ 4/3 · 3.14 · 1728

V ≈ 7,234.6

Hence, the volume of the soccer ball is about 7,234.6 cubic cm.

**Example 2 :**

Jose measures the diameter of a ball as 14 inches. How many cubic inches of air can the ball hold, to the nearest tenth ? Use the approximate of value of ∏, that is 3.14.

**Solution : **

**Step 1 : **

To know how much air the ball can hold, we have to find the volume of the ball.

Because a ball is in the shape of sphere, we can use the formula of volume of a sphere to find volume of a ball.

Write the formula to find volume of a sphere.

V = 4/3 · ∏r³ ----- (1)

**Step 2 : **

To find the volume, we need the radius of the sphere. But, the diameter is given, that is 14 cm. So, find the radius.

r = diameter / 2

r = 14/2

r = 7

**Step 3 : **

Plug ∏ ≈ 3.14 and r = 7 in (1).

V ≈ 4/3 · 3.14 · 7³

Simplify.

V ≈ 4/3 · 3.14 · 343

V ≈ 1,436

Hence, the ball can hold about 1,436 cubic inches.

**Example 3 :**

Air is leaking from a spherical-shaped advertising balloon at the rate of 26 cubic feet per minute. If the radius of the ball is 7 feet, how long would it take for the balloon to empty fully ? Round your answer to the nearest minute. Use the approximate of value of ∏, that is 3.14.

**Solution : **

**Step 1 : **

To know how long it would take for the balloon to empty fully, first we have to find the volume of air in the ballon.

**Because the balloon is in the shape of sphere, we can use the formula of volume of a sphere to find volume of air in the balloon. **

Write the formula to find volume of a sphere.

V = 4/3 · ∏r**²**h ----- (1)

**Step 2 : **

Plug ∏ ≈ 3.14 and r = 7 in (1).

V ≈ 4/3 · 3.14 · 7³

Simplify.

V ≈ 4/3 · 3.14 · 343

V ≈ 1,436

So, the volume of air in the balloon is about 1,436 feet.

**Step 3 : **

Air is leaking from the balloon at the rate of 26 cubic feet per minute.

To know how long it would take for the balloon to empty 1,426 cubic feet of air, divide 1,426 by 26.

= 1,426 / 26

= 54.8 ≈ 55

Hence, it would take about 55 minutes for the balloon to empty fully.

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