Question

2. A convenience store sells popsicles that come in two different shapes and are the same price. One Popsicle comes in the shape of an upside-down square-based pyramid that has a base length of 7cm and a depth of 11 cm. The other Popsicle is in the shape of a triangular prism. The prism is made up of an equilateral triangle with a side length of 9 cm and a height of 12cm. Which shape of Popsicle should you buy?

Answer 1

To determine which popsicle to buy, we need to calculate the volume of each shape.

### 1. Volume of Upside-Down Square-Based Pyramid

The formula for the volume \( V \) of a pyramid is given by:

\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

For the upside-down square-based pyramid:

- Base length \( b = 7 \) cm
- Depth (height) \( h = 11 \) cm

First, we calculate the base area:

\[
\text{Base Area} = b^2 = 7^2 = 49 \, \text{cm}^2
\]

Now we can calculate the volume:

\[
V = \frac{1}{3} \times 49 \times 11 = \frac{539}{3} \approx 179.67 \, \text{cm}^3
\]

### 2. Volume of Triangular Prism

The formula for the volume \( V \) of a prism is given by:

\[
V = \text{Base Area} \times \text{Height}
\]

For the triangular prism:

- Side length of the equilateral triangle \( s = 9 \) cm
- Height of the prism \( h = 12 \) cm

First, we calculate the base area of the equilateral triangle using the formula:

\[
\text{Base Area} = \frac{\sqrt{3}}{4} s^2
\]

Substituting in the side length:

\[
\text{Base Area} = \frac{\sqrt{3}}{4} \times 9^2 = \frac{\sqrt{3}}{4} \times 81 = \frac{81\sqrt{3}}{4} \, \text{cm}^2
\]

Now we can calculate the volume of the triangular prism:

\[
V = \text{Base Area} \times \text{Height} = \frac{81\sqrt{3}}{4} \times 12 = \frac{972\sqrt{3}}{4} = 243\sqrt{3} \, \text{cm}^3
\]

To get an approximate numerical value of \( 243\sqrt{3} \):

\[
\sqrt{3} \approx 1.732
\]

So,

\[
243\sqrt{3} \approx 243 \times 1.732 \approx 421.57 \, \text{cm}^3
\]

### Conclusion

- Volume of the upside-down square-based pyramid popsicle: \( \approx 179.67 \, \text{cm}^3 \)
- Volume of the triangular prism popsicle: \( \approx 421.57 \, \text{cm}^3 \)

Since both popsicles are the same price, you should buy the **triangular prism popsicle**, as it has a larger volume.