Answer: D
Explanation:
To find the perimeter of the rectangle, we need to know both the length and width of the rectangle. Since we are given the expression for the area, which is A(x) = x(14 – x), we can use this expression to find the length of the rectangle that corresponds to a given width x.
The area of a rectangle is equal to its length multiplied by its width. So we have:
A(x) = x(14 – x)
We can rewrite this equation as a quadratic equation in standard form:
A(x) = -x^2 + 14x
To find the maximum value of A(x), we can use the formula for the x-coordinate of the vertex of a quadratic function:
x = -b / 2a
where a = -1 and b = 14 in this case.
x = -14 / 2(-1) = 7
So the maximum area occurs when the width of the rectangle is 7.
To find the length of the rectangle, we can substitute x = 7 into the expression for the area:
A(7) = 7(14 – 7) = 49
So the length of the rectangle is 49/7 = 7.
Now we can find the perimeter of the rectangle by using the formula:
P = 2(l + w)
where l is the length and w is the width.
Substituting l = 7 and w = x, we get:
P = 2(7 + x)
So the correct answer is (D) 4x + 28.