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A right cone with a radius of 4 inches and a square pyramid both have a slant height of 5 inches. Both solids have the same surface area. Find the length of a base edge of the pyramid. Round your answer to the nearest hundredth.

User Efie
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Final answer:

To find the length of a base edge of the pyramid, we equate the surface area formulas of the cone and pyramid. This results in a quadratic equation that is solved to find the length. The length of the base edge is approximately 7.94 inches.

Step-by-step explanation:

To find the length of a base edge of the pyramid, we need to compare the surface areas of the cone and the pyramid. The formula for the surface area of a cone is:

A = πr² + πrl, where r is the radius and l is the slant height.

The formula for the surface area of a square pyramid is:

A = 2lw + l², where l is the length of the base edge and w is the slant height.

Since both solids have the same surface area, we can equate the two formulas:

πr² + πrl = 2lw + l².

Substituting the given values, we have:

π(4²) + π(4)(5) = 2l(5) + l².

Simplifying this equation, we get:

π(16) + π(20) = 10l + l².

Combining like terms, we have:

16π + 20π = 10l + l².

36π = 10l + l².

Rearranging the equation and setting it equal to zero, we have:

l² + 10l - 36π = 0.

Using the quadratic formula, we can solve for l:

l = (-b ± √(b² - 4ac)) / (2a), where a = 1, b = 10, and c = -36π.

Calculating the discriminant, we have:

b² - 4ac = 100 - 4(-36π) = 400 + 144π.

Taking the square root, we get:

√(400 + 144π) ≈ 24.87.

Substituting this value into the quadratic formula, we have:

l ≈ (-10 ± 24.87) / 2.

Taking the positive root, the length of the base edge of the pyramid is approximately:

l ≈ (24.87 - 10) / 2 ≈ 7.94 inches.

User Daniel Varela
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7.7k points
3 votes
The surface area of the cone is given by:
SA=πrl
Thus the surface area of the cone will be:
SA=π×4×5
SA=20π=63.932

Surface area of the pyramid is given by:
SA=1/2×4×(b×l)
thus the base length will be:
63.932=10l
thus
l=6.3932 in

User Eldar
by
7.9k points

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