Question

alicia has molds to make ice cubes in the shapes of two pyramids that have the same height but different bases. they are shown below. alicia says the volume of ice in shape 1, the rectangular pyramid, is three times the volume of ice in shape 2, the triangular pyramid. is alicia correct? use the formula below to determine the answer and use the drop - down menus to explain. $v = \frac{1}{3}bh$ ice molds (diagrams of shape 1: a rectangular pyramid with base length 4 cm, base width 3 cm, height 5 cm; shape 2: a triangular pyramid with base base 4 cm, base height 3 cm, height 5 cm)

Explanation:

Step1: Calculate Volume of Shape 1

Shape 1 is a rectangular pyramid. The base area \( B \) of a rectangle is length \( \times \) width. Here, length = 4 cm, width = 3 cm, so \( B_1 = 4 \times 3 = 12 \, \text{cm}^2 \). Height \( h = 5 \, \text{cm} \). Using the volume formula \( V = \frac{1}{3}Bh \), \( V_1 = \frac{1}{3} \times 12 \times 5 \).
\( V_1 = \frac{1}{3} \times 60 = 20 \, \text{cm}^3 \)

Step2: Calculate Volume of Shape 2

Shape 2 is a triangular pyramid. The base area \( B \) of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). Here, base = 4 cm, height = 3 cm, so \( B_2 = \frac{1}{2} \times 4 \times 3 = 6 \, \text{cm}^2 \). Height \( h = 5 \, \text{cm} \). Using the volume formula \( V = \frac{1}{3}Bh \), \( V_2 = \frac{1}{3} \times 6 \times 5 \).
\( V_2 = \frac{1}{3} \times 30 = 10 \, \text{cm}^3 \)

Step3: Compare Volumes

Now, we check the ratio of \( V_1 \) to \( V_2 \). \( \frac{V_1}{V_2} = \frac{20}{10} = 2 \). So \( V_1 \) is 2 times \( V_2 \), not 3 times.

Answer:

Alicia is not correct. The volume of the rectangular pyramid (Shape 1) is \( 20 \, \text{cm}^3 \) and the volume of the triangular pyramid (Shape 2) is \( 10 \, \text{cm}^3 \), so the volume of Shape 1 is 2 times the volume of Shape 2, not 3 times.