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Problem 10

# For a rectangle with area 100 to have the smallest perimeter, what dimensions should it have?

Expert verified
The dimensions of the rectangle with an area of 100 that yield the smallest perimeter are 10 × 10, with a perimeter of 40.
See the step by step solution

## Step 1: Calculate the possible dimensions of the rectangle with an area of 100.

We know that the area of a rectangle is calculated using the formula: $$A = l × w$$. We are given the area as 100, so we can write the equation as: $$100 = l × w$$. We need to find integer values for l and w that multiply to 100. Possible integer dimensions include: 1 × 100 2 × 50 4 × 25 5 × 20 10 × 10

## Step 2: Calculate the perimeter for each possible dimension set.

For each pair of dimensions (l, w), calculate the perimeter using the formula: $$P = 2 × l + 2 × w$$. 1. For dimensions 1 × 100: $$P = 2 × 1 + 2 × 100 = 2 + 200 = 202$$ 2. For dimensions 2 × 50: $$P = 2 × 2 + 2 × 50 = 4 + 100 = 104$$ 3. For dimensions 4 × 25: $$P = 2 × 4 + 2 × 25 = 8 + 50 = 58$$ 4. For dimensions 5 × 20: $$P = 2 × 5 + 2 × 20 = 10 + 40 = 50$$ 5. For dimensions 10 × 10: $$P = 2 × 10 + 2 × 10 = 20 + 20 = 40$$

## Step 3: Determine the dimensions with the smallest perimeter.

Now that we have the perimeters for each possible dimension set, we can compare them to find the smallest one. From our calculations, we can see that the rectangle with dimensions 10 × 10 has the smallest perimeter of 40. Therefore, the dimensions of the rectangle with an area of 100 that yield the smallest perimeter are 10 × 10.

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