Question

cam has 46 m of fencing to enclose a meditation space on the grounds of his local hospital. he has decided that the meditation space should be rectangular, with fencing on only three sides. what dimensions will give the patients the maximum amount of meditation space?

Answer 1

Cam can maximize the meditation space by creating a rectangle with dimensions 23 meters in length and 11.5 meters in width, using the available 46 meters of fencing to enclose the space on three sides.

Step-by-step explanation:

The student is tasked with determining the dimensions that maximize the area of a rectangular meditation space using a fixed amount of fencing.

In this case, Cam has 46 meters of fencing and needs to enclose a space with three sides, which means the fourth side is already existing, for example, a wall. If we denote the length of the rectangle by L and the width by W, and since only three sides need fencing, the perimeter P that Cam can use is 2W + L = 46 meters.

To maximize the area A of the rectangle, which is A = L * W, we can express L as L = 46 - 2W and substitute this into the area formula to get A = W * (46 - 2W). To find the maximum area, we set the derivative of A with respect to W to zero, A' = 46 - 4W = 0. Solving for W, we get W = 11.5 meters.

Substituting back into the perimeter equation, we find the length L = 46 - 2(11.5) = 23 meters. Therefore, the dimensions that give the maximum area are a length of 23 meters and a width of 11.5 meters.