Justin wants to use 188 ft of fencing to fence off the greatest possible rectangular area for a garden. What dimensions should he use? What will be the area of the garden?
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A. 89 ´ 99 ; 8811 ft
B. 92 ´ 96 ; 8832 ft
C. 94 ´ 94 ; 8836 ft
D. 93 ´ 95 ; 8835 ft
The first two numbers are the length and width of the garden, and in every case, they add up to 188; The third number is the area. The one with the greatest area is C. (94 * 94 = 8836)
Identify the expression as a numerical expression or a variable expression. For a variable expression, name the variable. 1 × 12
To find the dimensions that would result in the greatest possible rectangular area, we need to consider the relationship between the perimeter and the area of a rectangle.
We are given that Justin has 188 ft of fencing, so the perimeter of the rectangle would be 188 ft.
The formula for the perimeter of a rectangle is: P = 2L + 2W,
where L represents the length and W represents the width of the rectangle.
We can rearrange the formula to solve for one of the variables. Let's rearrange it to solve for L:
L = (P - 2W) / 2
Now, let's substitute the given perimeter value into the formula:
L = (188 - 2W) / 2
To maximize the area of the rectangle, we need to find the dimensions that will result in the maximum value of LW (length times width).
We can substitute the expression for L into the formula for LW:
A = LW = (188 - 2W) / 2 * W
A = (188W - 2W^2) / 2
A = 94W - W^2
To find the maximum area, we can take the derivative of the area formula with respect to W and set it equal to zero. Then, solve for W.
A' = 94 - 2W = 0
2W = 94
W = 47
Now that we have the value of W, we can substitute it back into the formula for L to find the length:
L = (188 - 2(47)) / 2
L = (188 - 94) / 2
L = 94 / 2
L = 47
So, the dimensions that would result in the greatest possible rectangular area are 47 ft by 47 ft.
The area of the garden would be:
Area = Length * Width
Area = 47 ft * 47 ft
Area = 2209 ft^2
Therefore, the correct answer is:
C. 94 ´ 94 ; 8836 ft