An oatmeal box is shaped like a cylinder with 2 inch radius and 9 inch height. Use a net to find the surface area S of the oatmeal box to the nearest tenth. Then find the number of square feet of cardboard needed for 1,500 oatmeal boxes. Round your answer to the nearest whole number.
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To find the surface area (S) of the oatmeal box, we need to calculate the area of the circular top and bottom, as well as the lateral surface area of the cylinder.
1. Area of the circular top and bottom:
The formula to find the area of a circle is A = π * r^2, where r is the radius.
Plugging in the given radius of 2 inches, we have:
A = π * (2^2)
A = π * 4
A ≈ 12.57 square inches
2. Lateral surface area of the cylinder:
The formula to find the lateral surface area of a cylinder is A = 2π * r * h, where r is the radius and h is the height.
Plugging in the given radius of 2 inches and height of 9 inches, we have:
A = 2π * 2 * 9
A = 36π
A ≈ 113.10 square inches
3. Total surface area (S) of the oatmeal box:
S = 2 * area of the circular top and bottom + lateral surface area of the cylinder
S = 2 * 12.57 + 113.10
S ≈ 138.24 square inches
To find the number of square feet of cardboard needed for 1,500 oatmeal boxes, we need to convert the surface area to square feet and then multiply by the number of boxes.
4. Converting to square feet:
Since 1 square foot is equal to 144 square inches, we divide the surface area by 144 to get the surface area in square feet.
Surface area in square feet = 138.24 / 144
Surface area in square feet ≈ 0.96 square feet
5. Multiplying by the number of boxes:
To find the total square footage needed for 1,500 boxes, we multiply the surface area in square feet by 1,500.
Total square footage = 0.96 * 1,500
Total square footage ≈ 1,440 square feet
Rounded to the nearest whole number, approximately 1,440 square feet of cardboard is needed for 1,500 oatmeal boxes.
To find the surface area (S) of the oatmeal box, we need to first find the area of the curved surface (lateral area) and the area of the top and bottom circles.
1. Curved Surface Area (Lateral Area):
The curved surface of the oatmeal box is a rectangle when it is unrolled, so we need to find the area of that rectangle.
The height of the rectangle is the same as the height of the oatmeal box, which is given as 9 inches.
To find the length of the rectangle, we need to find the circumference of the circle, which is the top/bottom of the box.
Circumference = 2πr, where r is the radius (given as 2 inches)
Circumference = 2 * 3.14 * 2 = 12.56 inches
So, the length of the rectangle (which is the height of the oatmeal box) is 9 inches.
The width of the rectangle (which is the circumference of the top/bottom circle) is 12.56 inches.
Curved Surface Area = length * width = 9 * 12.56 = 113.04 square inches
2. Area of the Top and Bottom Circles:
Each circle has a radius of 2 inches.
Area of a circle = πr^2 = 3.14 * (2)^2 = 3.14 * 4 = 12.56 square inches
Since we have two circles (top and bottom), the total area is 2 * 12.56 = 25.12 square inches.
Now, to find the total surface area, we add the curved surface area to the area of the top and bottom circles.
Surface Area (S) = Curved Surface Area + 2 * Area of Circles = 113.04 + 25.12 = 138.16 square inches
To find the number of square feet of cardboard needed for 1,500 oatmeal boxes, we multiply the surface area of one box (138.16 square inches) by 1,500.
Total square inches of cardboard = Surface Area * 1,500 = 138.16 * 1,500 = 207,240 square inches
To convert square inches to square feet, divide by 144 (since there are 144 square inches in a square foot).
Total square feet of cardboard = Total square inches of cardboard / 144 = 207,240 / 144 ≈ 1,440 square feet
Rounded to the nearest whole number, the number of square feet of cardboard needed for 1,500 oatmeal boxes is 1,440 square feet.
S = top + bottom + sides
top = pi r^2
bottom = pi r^2
sides = 2 pi r h
that will be in square inches (in^2)
there are 144 square inches per square foot
so
S * 1500 / 144