. You are making conical party hats with a height of 5.5 inches and a diameter of 3.5 inches. About how much cardboard is needed for each hat? Round your answer to the nearest hundredth.

Question

.. You are making conical party hats with a height of 5.5 inches and a diameter of 3.5 inches. About how much cardboard is needed for each hat?Round your answer to the nearest hundredth.

Answer 1

You are only looking for the later surface of the cone and the formula is:

LA =π•r•λ

Where:

r= radius

λ= slant height

if h = height of the cone

∴ λ² = r² + h²

because radius and height and slant height form a right triangle.

Given:

h= 5.5 inches

r = diameter/2 = 3.5 / 2 = 1.75 inches

λ² = (1.75)² + (5.5)²

λ² = 33.3125

λ ≈ 5.77

LA = π (1.75)•(5.77) ≈ 31.73 in²

*When you use your calculator, make sure you don't round off λ. You round off only on the final answer to make it accurate.

Answer 2

A party hat has NO base; therefore, πr² is not necessary. What we are looking at then - unfolded - looks similar to the pattern made by a windshield wiper on a dirty window.

The linear length of the "seam" can be called L. We calculate L with the Pythagorean theorem because L is a hypotenuse. L = ( r² + H² )^1/2, where r = D/2 = 1.75 inches, and H = 5.5 inches.

The lateral surface area of a cone is πrL. Ergo, our answer is roughly 31.73 squared inches.