Answer:
50 ft. × 50 ft.
Explanation:
Let's assume that the rectangular exhibit has a length of 'L' and a width of 'W'. The area of the exhibit is given as 2500 square feet, so we have:
L × W = 2500
We want to change the dimensions of the exhibit to minimize the amount of fencing needed while keeping the same area. The amount of fencing needed is directly proportional to the perimeter of the exhibit. The perimeter of a rectangular exhibit is given as:
P = 2L + 2W
We want to minimize P while keeping the area constant. One way to do this is to use the area formula to express one of the variables in terms of the other and then substitute it in the perimeter formula.
From the area formula, we have:
L × W = 2500
W = 2500 / L
Substituting W in the perimeter formula, we get:
P = 2L + 2(2500 / L)
Simplifying, we get:
P = 2L + 5000/L
To minimize P, we can take the derivative of P with respect to L and set it to zero:
dP/dL = 2 - 5000/L² = 0
Solving for L, we get:
L² = 2500
L = 50
Substituting L in the equation for W, we get:
W = 2500 / L = 50
Therefore, the new dimensions of the rectangular giraffe exhibit should be 50 feet by 50 feet to minimize the amount of fencing needed while keeping the same area.