Questions: A garden shop sells 12'' by 12'' square stepping stones for the same price as 18'' round stones. If all stepping stones are the same thickness, which option gives the most rock for the money? The area of a square stepping stone is in. ² and the area of a round stepping stone is in. ², so the option that gives the most rock for the money is the stepping stones.

Solution Steps

To determine which stepping stone option gives the most rock for the money, we need to calculate the area of both the square and round stepping stones. The area of a square stepping stone can be found using the formula for the area of a square, \( \text{side}^2 \). The area of a round stepping stone can be calculated using the formula for the area of a circle, \( \pi \times \text{radius}^2 \). Compare the two areas to determine which option provides more area for the same price.

The area \( A_s \) of a square stepping stone with a side length of \( 12 \) inches is calculated as follows: \[ A_s = \text{side}^2 = 12^2 = 144 \, \text{in}^2 \]

The area \( A_r \) of a round stepping stone with a diameter of \( 18 \) inches is calculated using the radius \( r \): \[ r = \frac{18}{2} = 9 \, \text{in} \] Thus, the area is: \[ A_r = \pi r^2 = \pi (9^2) = \pi \times 81 \approx 254.5 \, \text{in}^2 \]

Now we compare the two areas:

• Area of square stepping stone: \( 144 \, \text{in}^2 \)
• Area of round stepping stone: \( 254.5 \, \text{in}^2 \)

Since \( 254.5 > 144 \), the round stepping stone provides more area for the same price.

Final Answer