the perimeter of a rectangle is 300 feet. what is the width of the rectangle of maximun area?
The maximum area would make this rectangle a square.
To find the width of the rectangle that would result in the maximum area, we need to use calculus.
Let's denote the width of the rectangle as "w" and the length as "l". The formula for the perimeter of a rectangle is given by P = 2w + 2l. Given that the perimeter is 300 feet, we can write the equation as 2w + 2l = 300.
To find the width that maximizes the area, we need to maximize the area function, which is given by A = w * l. We can rewrite this equation in terms of "w" by solving the original equation for "l". From 2w + 2l = 300, we can isolate "l" as l = (300 - 2w) / 2, which simplifies to l = 150 - w.
Now we can substitute this expression for "l" into the area formula: A = w * (150 - w). Expanding the expression gives A = 150w - w^2.
To find the width that maximizes the area, we need to find the values of "w" where the derivative of the area function with respect to "w" (dA/dw) is equal to zero.
Let's find the derivative of A with respect to w: dA/dw = 150 - 2w.
Setting dA/dw equal to zero, we have 150 - 2w = 0. Solving for "w" gives w = 75.
Therefore, the width of the rectangle that results in the maximum area is 75 feet.