A gardener has 100 meters of fencing to enclose two adjacent rectangular gardens, as shown in the figure.(a) Write the area of the enclosed region as a function of x.A(x) =
Question
A gardener has 100 meters of fencing to enclose two adjacent rectangular gardens, as shown in the figure.(a) Write the area of the enclosed region as a function of x.A(x) =
Solution
Given that the gardener has 100 meters of fencing for two adjacent rectangular gardens, we can denote the length of the rectangle as x and the width as y.
Since the two gardens are adjacent and share a common length, the total fencing is used for 3 lengths and 2 widths. Therefore, we can write the equation for the perimeter as:
3x + 2y = 100
We can solve this equation for y to express it as a function of x:
y = (100 - 3x) / 2
The area A of a rectangle is given by the product of its length and width, so the total area of the two rectangles (which we'll call A(x)) is:
A(x) = x * y = x * (100 - 3x) / 2 = 50x - 1.5x^2
So, the area of the enclosed region as a function of x is A(x) = 50x - 1.5x^2.
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