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A variable rectangle has a constant perimeter of 20 cm. Find the lengths of the sides when the area is a maximum?

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Final answer:

To find the lengths of the sides when the area is a maximum, we can use the formulas for the perimeter and area of a rectangle. By solving the equations and finding the vertex of a quadratic equation, we can determine the lengths to be 5 cm each.

Step-by-step explanation:

The perimeter of a rectangle is given by the formula P = 2L + 2W, where P is the perimeter, L is the length, and W is the width. In this case, we are given that the perimeter is 20 cm, so we can write the equation as 2L + 2W = 20.

To find the lengths of the sides when the area is a maximum, we need to express the area in terms of a single variable. The area of a rectangle is given by the formula A = LW. We can solve the perimeter equation for one variable (L or W) and substitute it into the area equation.

Let's solve the perimeter equation for L:

L = (20 - 2W) / 2

Now substitute this value of L into the area equation:

A = (20 - 2W)W = 20W - 2W^2

To find the maximum area, we can find the vertex of the quadratic equation. The x-coordinate of the vertex is given by x = -b / 2a, where a = -2 and b = 20. Plugging in these values, we get:

W = -20 / (2*(-2)) = 5

Substituting this value back into the perimeter equation, we can find the length:

L = (20 - 2(5)) / 2 = 5

Therefore, the lengths of the sides when the area is a maximum are 5 cm and 5 cm.

User Knyu
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