suppose you have 54 feet of fencing to enclose a rectangle dog pen . the function a= 27x- x^2 , where x = width , gives you the area of the dog pen in square feet . what width gives you the maximum area ? what is the maximum area ? round to the nearest tenth as necessary .
the length of a garden is 2 m more than twice its width and its area is 24 m2, which of the following equations represents the give situationanswer
Suppose you have 54 feet of fencing to enclose a rectangular dog pen. The function , where x = width, gives you the area of the dog pen in square feet. What width gives you the maximum area? What is the maximum area? Round to the nearest tenth as necessary.
width = 13.5 ft; area = 546.8 ft2
width = 27 ft; area = 182.3 ft2
width = 27 ft; area = 391.5 ft2
width = 13.5 ft; area = 182.3 ft2
width =13.5
area=182.3
To find the width that gives you the maximum area of the dog pen, you need to first express the function in terms of the width.
The given function is: a = 27x - x^2
Next, you can use the fact that the sum of all four sides of the rectangle must equal the total length of the fence, which is 54 feet.
In a rectangle, the sum of the lengths of opposite sides must be twice the width. So, the perimeter of the rectangle can be expressed as: P = 2x + 2y, where x is the width and y is the length.
Given that the total length of the fence is 54 feet, we can write the equation as: 2x + 2y = 54
Since y represents the length, we can express it as: y = (54 - 2x) / 2, which simplifies to y = 27 - x
Now, substitute the value of y in the function for area with the value obtained for y from the equation of the perimeter:
A = x * (27 - x)
To find the maximum area, you can take the derivative of the function with respect to x, set it equal to zero, and solve for x.
dA/dx = 27 - 2x
Setting this derivative equal to zero and solving for x:
27 - 2x = 0
2x = 27
x = 27/2
x = 13.5
Thus, the width that gives you the maximum area is 13.5 feet.
To find the maximum area, substitute this value of x back into the function:
A = 13.5 * (27 - 13.5)
A = 13.5 * 13.5
A = 182.25 square feet
Therefore, the maximum area of the dog pen is approximately 182.25 square feet.