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 compare the volumes of the round and rectangular cake pans, we first need to calculate the volume of each pan.
For the round cake pan, the volume can be found using the formula for the volume of a cylinder: V = πr^2h, where V is the volume, r is the radius, and h is the height (or depth in this case). Since the diameter is 7 inches, the radius is 3.5 inches (7/2). The depth of the pan is 2 inches. Now, we can plug in the values:
V_round = π(3.5)^2(2) ≈ 3.14 × 12.25 × 2 ≈ 77.0 cubic inches.
For the rectangular cake pan, the volume can be found using the formula for the volume of a rectangular prism: V = lwh, where V is the volume, l is the length, w is the width, and h is the height (or depth in this case). The dimensions of the pan are 6 inches × 9 inches × 2 inches. Now, we can plug in the values:
V_rectangular = 6 × 9 × 2 = 108 cubic inches.
Comparing the volumes of the two pans, we see that the rectangular pan (108 cubic inches) has a larger volume than the round pan (77.0 cubic inches). Therefore, the rectangular cake pan has the larger volume.