Question

For a fence that uses 100 feet of fencing to be built around a rectangular garden, the area is given by A=50x-x^2, where x is the length, in feet. Find the dimensions of the garden with the maximum area. A. Length is 50, width is 10 B. Length is 25, width is 25 C. Length is 50, width is 25 D. Length is 30, width is 20

Answer 1

Answer:

Option B.

Step-by-step explanation:

Length of the fence = 100 feet.

Perimeter of the rectangular garden = 100 feet

2( length + width) = 100

Length + width = 50

Area of the garden is given by A = 50x - x²

If we draw the curve with area on y-axis and length on x axis, as shown in the picture attached,

Peak or vertex of the curve = [$-\frac{b}{2a}$, A(-$\frac{b}{2a}$)]

From the equation A = 50x - x²

a = (-1)

b = 50

So ($-\frac{b}{2a})=(-\frac{50}{2}$)

Therefore, for maximum area of the curve width = 25 feet

For x = 25,

Area of the garden = 50(25) - (25)²

A = 1250 - 625

A = 625 feet

Now area = length × width = 625

Length × 25 = 625

Length = $\frac{625}{25}$

Length = 25 feet

Therefore, dimensions of the rectangular field are 25 feet by 25 feet.

Option B. will be the answer.

Answer 2

Answer: length is 25, width is 25

Step-by-step explanation: