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?
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 cup, we need to calculate the lateral surface area of the cone.
The lateral surface area of a cone is given by the formula:
Lateral surface area = π * r * l
Where:
- π (pi) is a mathematical constant approximately equal to 3.14159
- r is the radius of the cone
- l is the slant height of the cone
In this case, the radius (r) is given as 3 inches. We need to find the slant height (l) of the cone, which can be calculated using the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by the height and the radius.
The Pythagorean theorem states that in a right triangle:
h^2 = r^2 + l^2
Where:
- h is the height
- r is the radius
- l is the slant height
In this case, the height (h) is given as 6 inches. Plugging these values into the equation, we can solve for the slant height:
6^2 = 3^2 + l^2
36 = 9 + l^2
l^2 = 36 - 9
l^2 = 27
l = sqrt(27)
l ≈ 5.2 inches
Now that we have the slant height, we can calculate the lateral surface area:
Lateral surface area = π * r * l
Lateral surface area ≈ 3.14159 * 3 * 5.2
Lateral surface area ≈ 49.08792 square inches
Therefore, approximately 49 square inches of paper are used to make the conical cup.