Two identical circles of largest possible area are to be cut from a 12in by 24in rectangular piece for stock. Find the amount of waste.
two circles of radius 6 in, any bigger will not fit.
circle area = 2 (pi 6^2) = 72 pi
rectanglearea = 12 * 24
waste = 12 *24 - 72 pi
Each circle has a diameter of 12 inches.
A = pi * r^2
A = 3.14 * 36
A = 113.04 sq. in.
2 * 113.04 = 226.08 sq. in.
(12 * 24) - 226.08 = ________ sq. in. waste
To find the amount of waste, we need to calculate the area of the rectangular piece that is not used after cutting out the circles.
1. Start by drawing the rectangular piece with dimensions 12in by 24in.
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| 24in |
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2. The two identical circles should be cut such that their centers are located exactly at the midpoints of the longer sides of the rectangle. This means that we need to divide the 24in side into two equal parts.
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| 12in | |
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| 12in |
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3. Since the circles are identical in size, each circle will have a radius of half the length of one side of the rectangle, which is 12in / 2 = 6in.
4. Calculate the area of each circle using the formula for the area of a circle: A = πr^2, where A is the area and r is the radius.
Area of one circle = π * (6in)^2 = π * 36in^2 = 36π in^2
5. Since we have two identical circles, the total area of both circles is 2 * 36π in^2 = 72π in^2.
6. Subtract the total area of the circles from the area of the rectangular piece to find the waste area:
Waste area = Rectangular area - Total area of circles
= 12in * 24in - 72π in^2
= 288in^2 - 72π in^2
So the amount of waste is 288in^2 - 72π in^2 or approximately 288in^2 - 226.195in^2 (using the approximation π ≈ 3.14159).
Simplifying further, the amount of waste is approximately 61.804in^2.
To find the amount of waste, we need to calculate the remaining area after cutting out the two circles.
Let's start by finding the radius of the circle. Since the circles have the largest possible area, they should touch opposite sides of the rectangle.
The length of the rectangle, 24 inches, will be the diameter of the circle. So, the radius will be half of the diameter, which is 24/2 = 12 inches.
Next, we need to find the area of each circle. The formula to calculate the area of a circle is:
Area = π * radius^2
Since the circles are identical, the area of each circle will be:
Area = π * (12 inches)^2
= 144π square inches
Now, to find the waste, we subtract the combined area of the two circles from the area of the rectangle:
Waste = (12 inches * 24 inches) - (2 * (144π square inches))
= 288 square inches - (288π square inches)
= 288 - 288π square inches
So, the amount of waste is 288 - 288π square inches, or approximately 73.155 square inches.