A manufacturer of canned fruit products is making a new label for its canned fruit salad. The can is 4 inches tall with a diameter of 3 inches. Determine the size of the label needed to cover the sides of the can to the nearest squared inch. How do I figure this out?
SA = 2 * pi * r * h
SA = 2 * 3.14 * 1.5 * 4
SA = 37.68 = 38 square inches
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To determine the size of the label needed to cover the sides of the can, you need to find the lateral surface area of the can. The lateral surface area is the sum of the areas of the curved surface around the can.
The formula to calculate the lateral surface area of a cylinder is:
Lateral Surface Area = 2πrh
Where:
- π is a mathematical constant equal to approximately 3.14159.
- r is the radius of the base of the cylinder (diameter divided by 2).
- h is the height of the cylinder.
In this case, the diameter of the can is 3 inches, so the radius would be 3/2 = 1.5 inches. The height of the can is 4 inches.
Let's calculate the lateral surface area:
Lateral Surface Area = 2π(1.5 inches)(4 inches)
Lateral Surface Area = 24π square inches
Now, we need to find the size of the label needed in terms of square inches. Since we are asked to round to the nearest square inch, we can use an approximation for π as 3.14.
Lateral Surface Area ≈ 24(3.14) square inches
Lateral Surface Area ≈ 75.36 square inches
Therefore, the label needed to cover the sides of the can would be approximately 75 square inches.
To determine the size of the label needed to cover the sides of the can, you first need to calculate the surface area of the can. The surface area is the sum of the areas of all the sides of the can.
To find the surface area, you can break down the can into three parts: the top and bottom circles, and the curved side surface.
1. Top and Bottom Circles:
The top and bottom circles have the same diameter as the can, which is 3 inches. Recall that the formula to find the area of a circle is A = πr^2, where r is the radius of the circle. Since the diameter is given, you can divide it by 2 to get the radius. Therefore, the area of each circle is A = π(3/2)^2.
2. Curved Side Surface:
The curved side surface of the can can be thought of as a rectangle that has been wrapped around the can. To find its area, you need to find the height and the length of the rectangle. The height of the rectangle is the same as the height of the can, which is 4 inches. The length of the rectangle is the circumference of the can, which can be found using the formula C = πd, where d is the diameter of the can. Therefore, the length of the rectangle is π(3).
Now, to calculate the surface area, you sum up the areas of all three parts: top circle + bottom circle + curved side surface.
Surface Area = 2(Area of top and bottom circles) + (Area of curved side surface)
Surface Area = 2(π(3/2)^2) + (π(3) * 4)
Once you calculate the surface area, round it to the nearest squared inch to determine the size of the label needed to cover the sides of the can.