Final answer:
To find the length and width of a rectangle that has a given perimeter and maximizes the area, we can set up an equation using the formula for the perimeter of a rectangle and solve for one variable in terms of the other.
Then, we can substitute this value into the formula for area and find the maximum.
In this case, the length and width of the rectangle that maximize the area with a perimeter of 76 meters are both 19 meters.
Step-by-step explanation:
In this case, the perimeter is given as 76 meters. We want to find the length and width that will maximize the area, so we can set up the equation 76 = 2L + 2W.
Since we want to find the maximum area, we can solve this equation for one variable and substitute it into the formula for area: A = LW.
Let's solve for L in terms of W:
76 - 2W = 2L
L = (76 - 2W)/2
= 38 - W
Now we can substitute this value of L into the formula for area and simplify:
A = (38 - W)W = 38W - W^2
Since we're looking for the maximum area, we can take the derivative of A with respect to W, set it equal to zero, and solve for W:
dA/dW = 38 - 2W
= 0
2W = 38
W = 19
Now we can substitute this value of W back into the equation for L:
L = 38 - 19
= 19
So the length and width of the rectangle that maximizes the area with a perimeter of 76 meters are 19 meters and 19 meters, respectively.