The diameter of a beach ball is 10 inches. How many cubic inches of air can the beach ball hold? Use 3.14 for [tex]$\pi$[/tex]. Round to the nearest tenth of a cubic inch.
Recall the formula Sphere [tex]$V=\frac{4}{3} \pi r^2$[/tex].
Answer (2)
Calculate the radius of the beach ball: r = 2 10 = 5 inches.
Substitute the radius and π into the volume formula: V = 3 4 ( 3.14 ) ( 5 3 ) .
Calculate the volume: V ≈ 523.333 cubic inches.
Round the volume to the nearest tenth: 523.3 in. 3
Explanation
Problem Analysis and Given Data We are given a beach ball with a diameter of 10 inches, and we need to find the volume of air it can hold. We are also given that we should use 3.14 as the value for π . The formula for the volume of a sphere is V = 3 4 π r 3 , where r is the radius of the sphere.
Calculate the Radius First, we need to find the radius of the beach ball. The radius is half of the diameter, so we have: r = 2 d = 2 10 = 5 So, the radius of the beach ball is 5 inches.
Calculate the Volume Now, we can substitute the radius r = 5 inches and π = 3.14 into the formula for the volume of a sphere: V = 3 4 π r 3 = 3 4 ( 3.14 ) ( 5 3 ) V = 3 4 ( 3.14 ) ( 125 ) V = 3 4 ( 392.5 ) V = 3 1570 V ≈ 523.333... So, the volume of the beach ball is approximately 523.333 cubic inches.
Round to the Nearest Tenth Finally, we need to round the volume to the nearest tenth of a cubic inch. Since the hundredths digit is 3, we round down to get: V ≈ 523.3 Therefore, the beach ball can hold approximately 523.3 cubic inches of air.
Examples
Understanding the volume of a sphere is useful in many real-world applications. For example, when designing spherical tanks for storing liquids or gases, engineers need to calculate the volume accurately to ensure the tanks can hold the required amount. Similarly, in sports, knowing the volume of a ball helps in determining its mass and aerodynamic properties, which are crucial for performance. Even in cooking, the volume of spherical fruits or vegetables can be estimated using this formula to help with portioning and recipe scaling. Imagine you're inflating a basketball and need to know how much air it will hold; this formula helps you determine that!
The volume of the beach ball can be calculated using the formula V = 3 4 π r 3 . After finding the radius as 5 inches and calculating the volume, we find it to be approximately 523.3 cubic inches when rounded to the nearest tenth. Therefore, the beach ball can hold around 523.3 cubic inches of air.
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