Answer: The area of the canvas required to make the tent is 420π square decimeters.
Explanation:
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To find the area of the canvas required to make the tent, we need to calculate the surface area of the cylinder and the surface area of the cone separately, and then add them together.
1. Surface area of the cylinder:
The formula for the surface area of a cylinder is given by 2πr1h, where r1 is the radius of the base and h is the height of the cylinder.
Given that the diameter of the base is 24 decimeters, we can find the radius (r1) by dividing the diameter by 2: r1 = 24/2 = 12 decimeters.
Substituting the values into the formula: 2π(12)(11) = 264π square decimeters.
2. Surface area of the cone:
The formula for the surface area of a cone is given by πr2l, where r2 is the radius of the base and l is the slant height of the cone.
We can find the slant height (l) using the Pythagorean theorem: l = √(h^2 + r2^2), where h is the height of the cone.
Given that the total height of the tent is 16 decimeters and the height of the cylinder is 11 decimeters, we can find the height of the cone (h2) by subtracting the height of the cylinder from the total height: h2 = 16 - 11 = 5 decimeters.
Substituting the values into the formula: l = √(5^2 + r2^2) = √(25 + r2^2).
The radius of the cone (r2) is half the diameter of the base, so r2 = 24/2 = 12 decimeters.
Substituting the values into the formula: l = √(25 + 12^2) = √(25 + 144) = √169 = 13 decimeters.
Finally, substituting the values into the surface area formula: π(12)(13) = 156π square decimeters.
3. Total surface area:
Adding the surface areas of the cylinder and cone together: 264π + 156π = 420π square decimeters.
Therefore, the area of the canvas required to make the tent is 420π square decimeters.