A cylindrical container of paint with a radius of 6 inches is 15 inches tall. If all of the surface area except the top are made of metal, how much metal is used to make the container? Assume the thickness of the material is negligable. Round to the nearest square inch.
If the top of the paint container is made of plastic, how much plastic is used to make the top? Assume the thickness of the plastic is negligable. Round to the nearest square inch.
First question:
the surface area would be the bottom circle + the rectangular sleeve
= π(6^2) + 2π(6)(15)
= 36π + 180π
= 216π square inches
Second question:
Unless I am missing something, but isn't that just the area of the circle which was 36π
To find the surface area of the cylindrical container, we need to calculate the surface area of the curved portion and the surface area of the bottom.
1. Surface area of the curved portion (lateral surface area):
The lateral surface area of a cylinder is given by the formula A = 2πrh, where A is the surface area, π is approximately 3.14, r is the radius, and h is the height.
In this case, the radius (r) is 6 inches and the height (h) is 15 inches. Plugging these values into the formula, we have:
A = 2 * 3.14 * 6 * 15
A ≈ 564.48 square inches
2. Surface area of the bottom (base):
The surface area of a circle is given by the formula A = πr^2, where A is the surface area and r is the radius.
In this case, the radius is 6 inches. Plugging this value into the formula, we have:
A = 3.14 * 6^2
A ≈ 113.04 square inches
3. Total surface area:
To find the total surface area, we add the surface area of the curved portion and the surface area of the bottom:
Total surface area ≈ 564.48 + 113.04 ≈ 677.52 square inches
Therefore, approximately 677.52 square inches of metal is used to make the cylindrical container.
To find the surface area of the cylindrical container, we need to consider three parts:
1. The top of the container, which is a circle.
2. The curved surface of the container, which is a rectangle when unrolled.
3. The bottom of the container, which is also a circle.
Step 1: Calculate the top and bottom surface areas (circles)
The formula for the area of a circle is A = π*r^2, where r is the radius.
Area of the top circle = π*(6 inches)^2
Area of the bottom circle = π*(6 inches)^2
Step 2: Calculate the curved surface area (unrolled rectangle)
The formula for the area of a rectangle is A = l * w, where l is the length and w is the width. In this case, the length will be equal to the circumference of the circle and the width will be the height of the cylindrical container.
The circumference of a circle is given by C = 2πr.
Circumference of the circle = 2π*(6 inches)
Now, the curved surface area is the rectangle obtained by unrolling the curved surface. The length of the rectangle will be the circumference and the width will be the height of the container.
Curved surface area = (2π*(6 inches)) * (15 inches)
Step 3: Add all the surface areas together
Total surface area of the cylindrical container = Area of the top circle + Area of the bottom circle + Curved surface area
Total surface area = (π*(6 inches)^2) + (π*(6 inches)^2) + (2π*(6 inches)*(15 inches))
Finally, calculate the value:
Total surface area ≈ 1075.33 square inches (rounded to the nearest square inch)
Therefore, approximately 1075 square inches of metal are used to make the container.