5. A recipe for lemon bars calls for a \( 9^{\prime \prime} \times 13^{\prime \prime} \) pan. How many \( 1^{\prime \prime} \times 1^{\prime \prime} \) bars only the cut? How many \( 2^{\prime \prime} \times 2^{\prime \prime} \) bars can be cut? How much is wasted if you serve only \( 2^{\prime \prime} \times 2^{\prime \prime} \) square bars?

To determine how many \( 1^{\prime \prime} \times 1^{\prime \prime} \) bars and \( 2^{\prime \prime} \times 2^{\prime \prime} \) bars can be cut from a \( 9^{\prime \prime} \times 13^{\prime \prime} \) pan, we can follow these steps: ### 1. Calculate the area of the pan: The area of the pan is given by: \[ \text{Area of the pan} = \text{length} \times \text{width} = 9 \times 13 = 117 \text{ square inches} \] ### 2. Calculate the number of \( 1^{\prime \prime} \times 1^{\prime \prime} \) bars: The area of each \( 1^{\prime \prime} \times 1^{\prime \prime} \) bar is: \[ \text{Area of a } 1^{\prime \prime} \text{ bar} = 1 \times 1 = 1 \text{ square inch} \] Thus, the number of \( 1^{\prime \prime} \times 1^{\prime \prime} \) bars that can be cut from the pan is: \[ \text{Number of } 1^{\prime \prime} \text{ bars} = \frac{\text{Area of the pan}}{\text{Area of a } 1^{\prime \prime} \text{ bar}} = \frac{117}{1} = 117 \] ### 3. Calculate the number of \( 2^{\prime \prime} \times 2^{\prime \prime} \) bars: The area of each \( 2^{\prime \prime} \times 2^{\prime \prime} \) bar is: \[ \text{Area of a } 2^{\prime \prime} \text{ bar} = 2 \times 2 = 4 \text{ square inches} \] To find the number of \( 2^{\prime \prime} \times 2^{\prime \prime} \) bars that can be cut, we need to consider how many can fit in the dimensions of the pan: - Along the 9-inch side: \( \frac{9}{2} = 4.5 \) (which means 4 full bars can fit) - Along the 13-inch side: \( \frac{13}{2} = 6.5 \) (which means 6 full bars can fit) Thus, the total number of \( 2^{\prime \prime} \times 2^{\prime \prime} \) bars that can be cut is: \[ \text{Number of } 2^{\prime \prime} \text{ bars} = 4 \times 6 = 24 \] ### 4. Calculate the wasted area: The total area used by the \( 2^{\prime \prime} \times 2^{\prime \prime} \) bars is: \[ \text{Area used} = \text{Number of } 2^{\prime \prime} \text{ bars} \times \text{Area of a } 2^{\prime \prime} \text{ bar} = 24 \times 4 = 96 \text{ square inches} \] The wasted area is then: \[ \text{Wasted area} = \text{Area of the pan} - \text{Area used} = 117 - 96 = 21 \text{ square inches} \] ### Summary: - Number of \( 1^{\prime \prime} \times 1^{\prime \prime} \) bars: **117** - Number of \( 2^{\prime \prime} \times 2^{\prime \prime} \) bars: **24** - Wasted area when serving \( 2^{\prime \prime} \times 2^{\prime \prime} \) bars: **21 square inches**