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.
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