Real-Life Math -- Solution
The key to solving this problem is to calculate the total surface
area of the hat, including the area of the crown.
The first step is
to calculate the total area of the hat as if it did not have a hole in the
middle (the crown) using the formula for calculating the area of a circle.
As
you can see from this diagram, the hat has a total diameter of 18 inches:
6 + 6 + 6 = 18
Since
r is half of the diameter, and pi is 3.14, all we have to do is plug in the
numbers.
Total area of hat = 3.14 x 92
Total
area of hat = 3.14 x 81
Total area of hat = 254.34 square inches
Now
you have to calculate the area of the "hole" (the crown).
Area of hole
= 3.14 x 32
Area of hole = 3.14 x 9
Area of hole = 28.26
square inches
Now subtract the area of the crown from the total
area of the hat to calculate how much material you will need for the rim.
254.34
- 28.26 = 226.08
So you need 226.08 square inches of felt for
the rim.
Now calculate the surface area of the crown.
The following
formula is used to calculate the surface area of a sphere.
And since the
crown is a semi-sphere, we can modify the formula above like this:
Area
of crown = 2 x 32 x 3.14
Area of crown = 2 x 9 x 3.14
Area
of crown = 56.52 square inches
The only thing left now is to add
the area of the crown to the area of the rim to calculate the total amount
of material needed.
Total amount = total area of rim + total area
of crown
Total amount = 226.08 square inches + 56.52 square inches
Total
amount = 282.6 square inches
Wayne Wichern is a milliner.
He says he uses math all the time, from drawing patterns to calculating how
much material he needs to setting his budget. "It is important," he says.
In fact, he says he wishes he had studied it more when he was in high school.