A joker's cap is in the form of right circular cone whose base radius is 7cm and height 24 cm. Find the area of the sheet required to make 10 such caps.
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We need to find the area of sheet required to make 10 joker's caps. Each cap is a right circular cone with base radius 7 centimeters and height 24 centimeters. The key insight is that we need the lateral surface area of the cone, since the base is open.
We have a problem about joker's caps shaped like right circular cones. Each cap has a base radius of 7 centimeters and a height of 24 centimeters. We need to find the total area of sheet material required to make 10 such caps.
To find the lateral surface area of the cone, we first need to calculate the slant height. Using the Pythagorean theorem, l squared equals r squared plus h squared. Substituting our values: l squared equals 7 squared plus 24 squared, which is 49 plus 576, equals 625. Therefore, l equals square root of 625, which is 25 centimeters.
The lateral surface area of a cone is given by the formula: A equals pi r l, where r is the base radius and l is the slant height. When we unfold the cone, we get a sector of a circle. Substituting our values of r equals 7 and l equals 25, we get A equals pi times 7 times 25, which equals 175 pi square centimeters.
Now we calculate the total area for 10 caps. For one cap, the lateral surface area is 175 pi square centimeters. For 10 caps, we multiply by 10 to get 1750 pi square centimeters. Converting to decimal form using pi approximately equals 3.14159, we get approximately 5497.8 square centimeters.
In conclusion, the area of sheet required to make 10 joker's caps is 1750 pi square centimeters, or approximately 5497.8 square centimeters. We solved this by first finding the slant height using the Pythagorean theorem, then applying the lateral surface area formula for a cone, and finally multiplying by 10 for the total number of caps.
The lateral surface area of a cone is given by the formula: A equals pi r l, where r is the base radius and l is the slant height. When we unfold the cone, we get a sector of a circle. Substituting our values of r equals 7 and l equals 25, we get A equals pi times 7 times 25, which equals 175 pi square centimeters.