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Difficulty: MediumCircular Area and Circumference Applications

A machinist is fabricating a circular metal part with a diameter of 1212 inches. To allow for bolts, four identical circular holes, each with a radius of 22 inches, are cut out of the part. What is the area of the remaining metal in square inches? (Use π=3.14\pi = 3.14)

Answer: 62.8 square inches

Answer

62.8
The remaining area is found by subtracting the total area of the four smaller circular holes from the area of the large circular part. The large part has a radius of 6 inches (half of the 12-inch diameter), giving an area of 113.04 square inches. Each of the four holes has a radius of 2 inches, yielding an area of 12.56 square inches per hole, or 50.24 square inches total. Subtracting 50.24 from 113.04 leaves 62.8 square inches.

Step-by-Step Solution

1
Find the radius of the large circular part.
Radius = 12 / 2 = 6 inches.
The area formula for a circle requires the radius, but the diameter was given.
2
Calculate the total area of the large circular part.
Area = 3.14 * 6^2 = 113.04 square inches.
This represents the total amount of metal before any holes are cut.
3
Calculate the area of one small bolt hole.
Area = 3.14 * 2^2 = 12.56 square inches.
Each hole is a smaller circle removed from the part.
4
Find the total area of all four bolt holes.
Total hole area = 12.56 * 4 = 50.24 square inches.
There are four identical holes that need to be accounted for.
5
Subtract the total hole area from the large part's area.
Remaining area = 113.04 - 50.24 = 62.8 square inches.
Subtracting the removed material gives the final remaining area of the metal part.

Key Concept

Calculating and subtracting circular areas
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Topics

Circular Area and Circumference Applications

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