Problem of the Week Archive
Custom Costume – October 24, 2016
Problems & Solutions
Harry wants to dress up as a wizard for Halloween. He decides to make his own costume and
designs a wizard hat to be a right circular cone. Harry measures the circumference of his head to
be 18 inches, and he wants the hat to have a height of 12 inches. He uses a sector of a circle of
radius r, shown in Figure 1, as a pattern to cut a piece of fabric, which he rolls into a cone and then
sews the two edges together to create a hat, shown in Figure 2. What is the radius r of the sector
Harry uses as a pattern? Express your answer as a decimal to the nearest tenth.
The height of the cone is measured perpendicular to its circular base. The radius of the base can be found from the
given circumference. The radius of the base is 18 ÷ 2π = 9/π inches. The height, the radius of the base and the
radius of the sector form a right triangle. Plugging into the Pythagorean Theorem, we have r2 = 122 + (9/π)2, so
r = √(144 + 81/π2) ≈ 12.3 inches.
Harry wants to know how much fabric he needs to make his hat. What is the area, in square inches,
of the sector he uses to create his hat? Express your answer as a decimal to the nearest tenth.
The fabric Harry uses is a sector of a circle whose radius is 12.3 inches. The arc length of the
sector is only 18 inches, compared to the circumference of the entire circle, which is 2 × π × 12.3
= 24.6π inches. We can multiply the area of the entire circle, which is π × 12.32 in2, by the ratio
18/24.6π = 3/4.1π to determine the area of the sector. We get (3/4.1π) × π × 12.32 = 110.7 in2.
Harry plans to use his new wizard hat to hold the candy he collects as he goes trick-or-treating
around his neighborhood. How many gallons of candy will Harry’s wizard hat hold, given that
231 in3 = 1 gallon? Express your answer as a decimal to the nearest tenth.
Substituting the dimensions of Harry’s hat into the formula for the volume of a right circular cone, we get a volume
of 1/3 × π × r2 × h = (1/3) × π × (9/π)2 × 12 = 324/π in3, which is equivalent to 324/π ÷ 231 ≈ 0.4 gallons.
Problem of the Week Archive
Custom Costume – October 24, 2016
Problems
Harry wants to dress up as a wizard for Halloween. He decides to make his own costume and
designs a wizard hat to be a right circular cone. Harry measures the circumference of his head to
be 18 inches, and he wants the hat to have a height of 12 inches. He uses a sector of a circle of
radius r, shown in Figure 1, as a pattern to cut a piece of fabric, which he rolls into a cone and then
sews the two edges together to create a hat, shown in Figure 2. What is the radius r of the sector
Harry uses as a pattern? Express your answer as a decimal to the nearest tenth.
Harry wants to know how much fabric he needs to make his hat. What is the area, in square inches,
of the sector he uses to create his hat? Express your answer as a decimal to the nearest tenth.
Harry plans to use his new wizard hat to hold the candy he collects as he goes trick-or-treating
around his neighborhood. How many gallons of candy will Harry’s wizard hat hold, given that
231 in3 = 1 gallon? Express your answer as a decimal to the nearest tenth.