Math 11 Quadratic functions max and min problems

Question

Determine the maximized area of a rectangle that has a perimeter equal to 56m^2 by creating and solving a quadratic equation. What is the length and width of this rectangle?

Answer 1

I am assuming you mean perimeter is 56 meters. It's the area that is in meters squared. This can be done as a calculus problem but also using algebra, so I will assume you want algebra. Once the equation is constructed, it's easy to see that you get the same answer with calculus. First, you have 2L + 2W = 56. Divide equation by two. L + W = 28.

What else do you know? L*W = Area. Substitute let's say for W, W = 28 - L from the first equation. You get

L * (28 - L) = Area. Multiply through. 28L - L² = Area. Rearrange, and you get L² -28L - Area = 0. This is your quadratic function. Area is your constant C, a = 1 and b = -28. Remember the formula for the x value of the vertex of a quadratic. That is -b/2a. Solve for this and you get L. Then solve for W.