A gardener has 40 feet of fencing with which to enclose a garden adjacent to a long existing wall. The gardener will use the wall for one side and the available fencing for the remaining three sides.
If the sides perpendicular to the wall have length x feet, which of the following (A, B, C, or D) represents the area A of the garden? (No explanation required) Choice:
A. A(x) = –2x^2 + 20x
B. A(x) = –2x^2 + 40x
C. A(x) = 2x^2 – 40x
D. A(x) = x^2 – 40x
I say it's B
and based on that
The parabola opens down? agree or not?
Vertex = (10,200)??
The maximum area is 200 sqft?
When the sides perpendicular to the wall have length x = 10ft.
and the side parallel to the wall has length 20ft.
Is this all correct and accurate? Thanks!
Yes, that is still accurate.
Houston Community College is planning to construct a rectangular parking lot on land
bordered on one side by a highway. The plan is to use 640 ft of fencing to fence off the
other three sides. What should the dimensions of the lot be if the enclosed area is to be a
maximum?
To find the correct answer choice, we need to set up an equation based on the information given.
Let's denote the length of the sides perpendicular to the wall as x feet. Since we have 40 feet of fencing available for the remaining three sides, the sum of the lengths of those three sides must be equal to 40.
Since the wall is one side of the garden and has no assigned length variable, we can ignore it when setting up the equation. Therefore, we have:
2x + side parallel to the wall = 40
Simplifying the equation, we have:
2x + side parallel to the wall = 40
2x + 20 = 40
2x = 20
x = 10
Therefore, the length of the sides perpendicular to the wall is 10 feet, and the length of the side parallel to the wall is 20 feet.
Let's now evaluate each answer choice:
A. A(x) = –2x^2 + 20x
Plugging in x = 10, we get A(10) = -2(10)^2 + 20(10) = -200 + 200 = 0. This answer does not make sense since the area cannot be zero when the sides have non-zero length.
B. A(x) = –2x^2 + 40x
Plugging in x = 10, we get A(10) = -2(10)^2 + 40(10) = -200 + 400 = 200. This answer choice matches the information given.
C. A(x) = 2x^2 – 40x
Plugging in x = 10, we get A(10) = 2(10)^2 - 40(10) = 200 - 400 = -200. This answer does not make sense since the area cannot be negative.
D. A(x) = x^2 – 40x
Plugging in x = 10, we get A(10) = (10)^2 - 40(10) = 100 - 400 = -300. This answer does not make sense since the area cannot be negative.
Therefore, the correct answer choice is B. A(x) = –2x^2 + 40x.
Regarding the additional questions:
1. The parabola defined by function A(x) = –2x^2 + 40x opens downward because the leading coefficient (-2) is negative.
2. The vertex of the parabola is at (10, 200), which means it corresponds to x = 10 and A(x) = 200. This indicates that the maximum area of the garden is indeed 200 sqft.
3. When the sides perpendicular to the wall have a length of x = 10ft and the side parallel to the wall has a length of 20ft, the maximum area of the garden is achieved, which is 200 sqft.
So, your answers are correct and accurate!