The diameter of a large lawn ornament in the shape of a sphere is 16 inches. What is the approximate volume of the ornament? Use 3.14 for [tex]$\pi$[/tex]. Round to the nearest tenth of a cubic inch.
Recall the formula Sphere [tex]$V=\frac{4}{3} \pi r^3$[/tex].
Answer (2)
Calculate the radius of the sphere: r = 2 16 = 8 inches.
Substitute the radius and π into the volume formula: V = 3 4 ( 3.14 ) ( 8 3 ) .
Calculate the volume: V ≈ 2143.57333 cubic inches.
Round the volume to the nearest tenth: V ≈ 2143.6 in. 3
Explanation
Problem Analysis and Given Data We are given a sphere-shaped lawn ornament with a diameter of 16 inches. Our goal is to find the approximate volume of this ornament, using 3.14 as the value for π and rounding our final answer to the nearest tenth of a cubic inch. We'll use the formula for the volume of a sphere, which 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 sphere. The radius is half of the diameter, so we have: r = 2 d = 2 16 = 8 Thus, the radius of the sphere is 8 inches.
Substitute Values into Volume Formula Now, we'll substitute the radius r = 8 inches and π = 3.14 into the volume formula: V = 3 4 π r 3 = 3 4 ( 3.14 ) ( 8 3 ) We know that 8 3 = 8 × 8 × 8 = 512 , so we have: V = 3 4 ( 3.14 ) ( 512 ) V = 3 4 × 3.14 × 512 V = 3 6430.72 V ≈ 2143.57333
Round to Nearest Tenth Finally, we need to round the volume to the nearest tenth of a cubic inch. Looking at the calculated volume V ≈ 2143.57333 , we see that the digit in the tenths place is 5, and the digit to the right of it is 7. Since 7 is greater than or equal to 5, we round up the 5 to a 6. Therefore, the approximate volume of the lawn ornament is 2143.6 cubic inches.
Final Answer The approximate volume of the lawn ornament is 2143.6 cubic inches.
Examples
Imagine you're designing inflatable beach balls. Knowing how to calculate the volume of a sphere helps you determine how much air is needed to fully inflate each ball. This ensures you use the right amount of material and air, optimizing production costs and creating a consistent product. Also, understanding sphere volumes is crucial in fields like pharmaceuticals, where precise dosages in spherical capsules are essential for effective medication.
The approximate volume of the sphere-shaped lawn ornament with a diameter of 16 inches is 2143.6 cubic inches. This is calculated using the volume formula for a sphere, finding the radius, and rounding to the nearest tenth. Hence, after all calculations, we find the volume to be 2143.6 cubic inches.
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