A conical paper cup for holding popcorn has a radius of 3 in. and height of 6 in. How much paper, to the nearest square inch, is used to make the cup?

To find the amount of paper used to make the conical cup, we need to calculate the lateral surface area of the cone.

First, let's find the slant height of the cone using the Pythagorean theorem. The slant height (l) is the hypotenuse of the right triangle formed by the height (h) and the radius (r) of the cone.

Using the formula of the Pythagorean theorem: c^2 = a^2 + b^2, and substituting the given values:
l^2 = 3^2 + 6^2
l^2 = 9 + 36
l^2 = 45
l ≈ √45
l ≈ 6.708 in (rounded to three decimal places)

Now that we have the slant height, we can calculate the lateral surface area (A) of a cone using the formula A = πrl, where π is a mathematical constant approximately equal to 3.14159.

Using the formula:
A = π × 3 × 6.708
A ≈ 63.585 square inches (rounded to three decimal places)

Therefore, approximately 63.585 square inches of paper are used to make the conical cup.