When baking a cake, you have a choice of the following pans: a round cake pan that is 2 inches deep and has a 7 inch diameter.
a 6 inch x9 inch rectangular cake pan that is 2 inches deep.
A. which of these pans has the larger value? Justify your answer.
B. Excluding the bottom of the cake, what is the surface area of each cake that will be covered by frosting? Justify your answer.
V (round) = π(r^2)h
V Rectangular = 6*9*2
A (round-bottom) = 2πrh + πr^2
A (rect.-bottom) = 6*9 + 2*2*9 + 2*2*6
A. To determine which pan has the larger value, we can compare their volumes. The volume of a round cake pan can be calculated using the formula V = πr^2h, where r is the radius of the pan and h is the height. In this case, the radius is half of the diameter, so r = 7/2 = 3.5 inches. Therefore, the volume of the round cake pan is V = π(3.5)^2(2) = 38.5π cubic inches.
The volume of a rectangular cake pan can be calculated using the formula V = lwh, where l is the length, w is the width, and h is the height. In this case, the length is 9 inches, the width is 6 inches, and the height is 2 inches. Therefore, the volume of the rectangular cake pan is V = (9)(6)(2) = 108 cubic inches.
Comparing the volumes, 38.5π < 108. Therefore, the rectangular cake pan has the larger value.
B. To determine the surface area of each cake that will be covered by frosting, we need to calculate the surface area of the top and sides of each pan.
For the round cake pan, the area of the top surface is A_top = πr^2 = π(3.5)^2 = 38.5π square inches. The area of the side surface is A_side = 2πrh = 2π(3.5)(2) = 14π square inches. Therefore, the total surface area of the round cake pan that will be covered by frosting is A_top + A_side = 38.5π + 14π = 52.5π square inches.
For the rectangular cake pan, the area of the top surface is A_top = lw = (9)(6) = 54 square inches. The area of the two longer sides is A_long = lh + lh = 2(9)(2) = 36 square inches. The area of the two shorter sides is A_short = lw + lw = 2(6)(2) = 24 square inches. Therefore, the total surface area of the rectangular cake pan that will be covered by frosting is A_top + A_long + A_short = 54 + 36 + 24 = 114 square inches.
Therefore, the surface area of the rectangular cake pan that will be covered by frosting is larger than the surface area of the round cake pan.
To determine which pan has the larger value, we can compare the volumes of the two pans. The volume of a three-dimensional object like a cake pan can be calculated by multiplying its width, length, and depth.
A. To find the volume of the round cake pan, we need to calculate the volume of a cylinder. The formula for the volume of a cylinder is V = π * r^2 * h, where π is a mathematical constant approximately equal to 3.14159, r is the radius of the circular base, and h is the height of the cylinder.
Given that the diameter of the round cake pan is 7 inches, the radius is half of that, so the radius is 7/2 = 3.5 inches. The depth of the cake pan is given as 2 inches.
Plugging these values into the formula, the volume of the round cake pan is V = 3.14159 * (3.5^2) * 2 = 38.484 cubic inches.
Next, let's find the volume of the rectangular cake pan. The formula for the volume of a rectangular prism is V = l * w * h, where l is the length, w is the width, and h is the height.
Given that the rectangular cake pan has dimensions of 6 inches by 9 inches and a depth of 2 inches, the volume is V = 6 * 9 * 2 = 108 cubic inches.
Comparing the two volumes, we see that the rectangular cake pan has a larger value of 108 cubic inches, whereas the round cake pan has a volume of 38.484 cubic inches. Therefore, the rectangular cake pan has the larger value.
B. To calculate the surface area of each cake that will be covered by frosting, we need to find the area of the top surface and the area of the sides for each pan.
Starting with the round cake pan, the area of the top surface is found using the formula for the area of a circle, A = π * r^2. Given a radius of 3.5 inches, the area of the top surface is A = 3.14159 * (3.5^2) = 38.484 square inches (which we already calculated as the volume of the round cake pan).
The area of the sides of the round cake pan can be found by multiplying the circumference of the base (the circle) with the height of the pan. The formula for the circumference of a circle is C = 2 * π * r.
Given a radius of 3.5 inches and a height of 2 inches, the area of the sides is A = (2 * 3.14159 * 3.5) * 2 = 43.982 square inches.
Adding the area of the top surface to the area of the sides, we get the total surface area of the round cake pan covered by frosting: 38.484 + 43.982 = 82.466 square inches.
Moving on to the rectangular cake pan, the area of the top surface is simply the product of its length and width: A = 6 * 9 = 54 square inches.
The area of the sides is found by multiplying the perimeter of the top surface (rectangle) with the height of the pan. The perimeter of a rectangle is given by P = 2 * (l + w).
For the rectangular cake pan, the perimeter is P = 2 * (6 + 9) = 30 inches. Multiplying this by the height of 2 inches, we get the area of the sides as A = 30 * 2 = 60 square inches.
Adding the area of the top surface to the area of the sides, we get the total surface area of the rectangular cake pan covered by frosting: 54 + 60 = 114 square inches.
Therefore, the rectangular cake pan has a larger surface area covered by frosting, with 114 square inches as compared to the round cake pan's surface area of 82.466 square inches.