# How do you find the dimensions of a rectangle whose area is 100 square meters and whose perimeter is a minimum?

By taking the derivative,

By testing some sample values,

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To find the dimensions of a rectangle with a given area of 100 square meters and a minimum perimeter, you would need to apply the concept of optimization. Let ( l ) represent the length of the rectangle and ( w ) represent the width. The formulas for area and perimeter of a rectangle are:

Area = Length ( \times ) Width Perimeter = 2( \times ) (Length + Width)

To minimize the perimeter while keeping the area constant, you can use the following steps:

- Express one of the dimensions in terms of the other using the area formula.
- Substitute the expression for one dimension into the formula for the perimeter.
- Find the derivative of the resulting expression with respect to the remaining variable.
- Set the derivative equal to zero and solve for the variable.
- Use the obtained value to find the dimensions.

Given the area ( A = 100 ) square meters:

- Express one dimension in terms of the other: ( l = \frac{100}{w} ).
- Substitute into the perimeter formula: ( P = 2(l + w) = 2\left(\frac{100}{w} + w\right) ).
- Find the derivative of the perimeter with respect to width: ( \frac{dP}{dw} = -\frac{200}{w^2} + 2 ).
- Set the derivative equal to zero and solve for width: ( -\frac{200}{w^2} + 2 = 0 ).
- Solve for ( w ): ( w^3 = 100 \Rightarrow w = \sqrt[3]{100} ).
- Find the corresponding length using the formula for length: ( l = \frac{100}{w} ).

Thus, the dimensions of the rectangle that minimize its perimeter are approximately ( w \approx 4.64 ) meters (width) and ( l \approx 21.54 ) meters (length).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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