The entire can needs to be painted, including the top and bottom. If the diameter of the lid (x) is 5 cm, and the height of the can (y) is 9 cm, what is the approximate total surface area that will need to be painted?
141.37 + 39.26 = 180.63
surface area
= 2π(2.5)^2 + 2π(2.5)(9)
= ---
To find the total surface area that needs to be painted, we need to calculate the areas of the sides, top, and bottom of the can separately, and then add them together.
1. Area of the top and bottom:
The top and bottom of the can are both circles. The formula for the area of a circle is A = πr^2, where r is the radius.
Given the diameter (x) of the lid is 5 cm, the radius (r) is half of the diameter, so r = x/2 = 5/2 = 2.5 cm.
The area of the top and bottom of the can is:
A_top_bottom = 2 * π * r^2 = 2 * π * (2.5)^2 ≈ 39.27 cm^2
2. Area of the side:
The side of the can is like a rectangle that has been rolled into a cylinder. The formula for the area of a rectangle is A = l * w, where l is the length and w is the width.
Given the height of the can (y) is 9 cm, the width of the rectangle is equal to the circumference (C) of the lid, which can be calculated using the formula C = πd, where d is the diameter.
The width of the rectangle (w) is: w = C = πx = π * 5 ≈ 15.71 cm
And the length (l) is equal to the height (y) of the can: l = y = 9 cm
The area of the side of the can is:
A_side = l * w = 9 * 15.71 ≈ 141.39 cm^2
3. Total surface area:
To find the total surface area, we add together the areas of the top, bottom, and side:
Total surface area = A_top_bottom + A_side ≈ 39.27 + 141.39 = 180.66 cm^2
Therefore, the approximate total surface area that will need to be painted is approximately 180.66 cm^2.
To find the total surface area that needs to be painted, we need to calculate the area of the lid, the area of the bottom, and the area of the curved surface of the can.
1. The area of the lid: The lid is in the shape of a circle, so we can calculate its area using the formula A = πr^2, where r is the radius of the lid. The diameter of the lid (x) is given as 5 cm, so the radius (r) is half of the diameter, which is 5/2 = 2.5 cm. Plugging these values into the formula, we get A_lid = π(2.5)^2.
2. The area of the bottom: Similar to the lid, the bottom of the can is also a circle. So, the area of the bottom will also be A_bottom = πr^2, where r is the radius of the bottom. Since the radius is the same as for the lid, the area of the bottom is also A_bottom = π(2.5)^2.
3. The area of the curved surface: The curved surface of the can is in the shape of a rectangle when "unwrapped." The formula to calculate the area of a rectangle is A = 2πrh, where r is the radius of the can, and h is the height of the can. We are given the radius (r = 2.5 cm) and the height (y = 9 cm), so the area of the curved surface is A_curved = 2π(2.5)(9).
To find the total surface area, we add together the areas of the lid, bottom, and curved surface: Total surface area = A_lid + A_bottom + A_curved.
Substituting the respective values, we can calculate the approximate total surface area that needs to be painted.