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 × 9 inch rectangular cake pan that is 2 inches deep
Which of these pans has the larger volume? Justify your answer.
To find the volume of each pan, we need to use the formula for the volume of a cylinder (V = πr^2h) for the round pan and the formula for the volume of a rectangular prism (V = lwh) for the rectangular pan.
Round pan:
radius = 7/2 = 3.5 inches
height = 2 inches
V = π(3.5)^2(2) = 24.46π cubic inches
Rectangular pan:
length = 9 inches
width = 6 inches
height = 2 inches
V = (9)(6)(2) = 108 cubic inches
Therefore, the rectangular pan has a larger volume than the round pan (108 cubic inches vs. 24.46π cubic inches).
To determine which pan has the larger volume, we can calculate the volume of each pan using the formula for the volume of a cylinder for the round pan and the formula for the volume of a rectangular prism for the rectangular pan.
Let's calculate the volume of the round pan first:
The formula for the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height.
In this case, the round pan has a diameter of 7 inches, so the radius is half of that, which is 7/2 = 3.5 inches. The height of the pan is 2 inches.
Plugging these values into the formula, we get:
V = π * (3.5)^2 * 2 ≈ 38.48 cubic inches
Now let's calculate the volume of the rectangular pan:
The formula for the volume of a rectangular prism is V = lwh, where V is the volume, l is the length, w is the width, and h is the height.
In this case, the rectangular pan has a length of 9 inches, a width of 6 inches, and a height of 2 inches.
Plugging these values into the formula, we get:
V = 9 * 6 * 2 = 108 cubic inches.
Comparing the two volumes, we can see that the rectangular pan has a larger volume of 108 cubic inches compared to the round pan's volume of approximately 38.48 cubic inches.
Therefore, the rectangular pan has the larger volume.