Find the dimension of a rectangle with perimeter 295 whose area is as large as possible
To find the dimensions of a rectangle with the largest possible area given a fixed perimeter, we can use the derivative to find the maximum.
Let's start by assigning variables. Let the length of the rectangle be L and the width be W.
We know that the perimeter of the rectangle is 295, so we can write the equation:
2L + 2W = 295
Simplifying this equation, we get:
L + W = 147.5
From the equation for the area of a rectangle, A = L * W, we can express L in terms of W:
L = 147.5 - W
Substituting this expression for L in the area equation, we get:
A = (147.5 - W) * W
Expanding the equation, we have:
A = 147.5W - W^2
To find the maximum area, we need to find the critical points of this function. Taking the derivative of A with respect to W:
dA/dW = 147.5 - 2W
Setting the derivative equal to zero and solving for W:
147.5 - 2W = 0
2W = 147.5
W = 73.75
Now, substitute this value of W back into the equation for L:
L = 147.5 - W
L = 147.5 - 73.75
L = 73.75
So, the dimensions of the rectangle with the largest possible area, given a perimeter of 295, are L = 73.75 and W = 73.75.