A rectangular piece of cardboard is 3 ft 11 in long and 2 ft 4 in wide. What is the maximum number of 5 in by 8 in rectangles that can be cut from it?

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UpStudy AI · Accepted Answer

To solve the problem, we need to follow these steps:

1. **Convert the dimensions of the cardboard from feet and inches to inches only.**
2. **Calculate the area of the cardboard.**
3. **Calculate the area of one of the smaller rectangles (5 in by 8 in).**
4. **Determine how many of the smaller rectangles can fit into the area of the cardboard.**

### Step 1: Convert dimensions to inches

- The length of the cardboard is 3 ft 11 in.
  - Convert feet to inches: $$3 \text{ ft} = 3 \times 12 = 36 \text{ in}$$
  - Total length in inches: $$36 \text{ in} + 11 \text{ in} = 47 \text{ in}$$

- The width of the cardboard is 2 ft 4 in.
  - Convert feet to inches: $$2 \text{ ft} = 2 \times 12 = 24 \text{ in}$$
  - Total width in inches: $$24 \text{ in} + 4 \text{ in} = 28 \text{ in}$$

### Step 2: Calculate the area of the cardboard

The area $$A_c$$ of the cardboard can be calculated using the formula:
$$
A_c = \text{length} \times \text{width} = 47 \text{ in} \times 28 \text{ in}
$$

### Step 3: Calculate the area of one of the smaller rectangles

The area $$A_r$$ of one of the smaller rectangles (5 in by 8 in) is:
$$
A_r = 5 \text{ in} \times 8 \text{ in}
$$

### Step 4: Determine how many rectangles can fit

To find the maximum number of rectangles that can fit, we divide the area of the cardboard by the area of one rectangle:
$$
\text{Number of rectangles} = \frac{A_c}{A_r}
$$

Now, let's perform the calculations.

1. Calculate $$A_c$$.
2. Calculate $$A_r$$.
3. Calculate the number of rectangles.

Let's proceed with these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$47\times 28$$
- step1: Multiply the numbers:
  $$1316$$
Calculate or simplify the expression $$ 5 * 8 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$5\times 8$$
- step1: Multiply the numbers:
  $$40$$
Calculate or simplify the expression $$ 1316 / 40 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1316}{40}$$
- step1: Reduce the fraction:
  $$\frac{329}{10}$$
Now that we have the necessary calculations:

1. The area of the cardboard $$A_c$$ is $$1316 \text{ in}^2$$.
2. The area of one of the smaller rectangles $$A_r$$ is $$40 \text{ in}^2$$.

### Step 4: Calculate the number of rectangles

To find the maximum number of rectangles that can fit, we divide the area of the cardboard by the area of one rectangle:
$$
\text{Number of rectangles} = \frac{1316}{40} = 32.9
$$

Since we can only cut whole rectangles, we take the integer part of this result. Therefore, the maximum number of 5 in by 8 in rectangles that can be cut from the cardboard is:

$$
\boxed{32}
$$
The maximum number of 5 in by 8 in rectangles that can be cut from the cardboard is 32.

A rectangular piece of cardboard is 3 ft 11 in long and 2 ft 4 in wide. What is the maximum number of 5 in by 8 in rectangles that can be cut from it?

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