Mary wants to build a fence around a rectangular area for a garden. She has 150 ft of fencing and she wants to leave 10 ft opening on side for a gate. In order to make a area of the garden. A maximum what should the dimensions of the garden will be?
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To find the dimensions of the garden that would result in the maximum area, we need to consider the given constraints. Mary has 150 ft of fencing, and she wants to leave a 10 ft opening for a gate.
Let's assume the length of the garden is L and the width of the garden is W. Since there are two equal lengths and two equal widths in a rectangle, we can express the perimeter of the garden in terms of L and W as:
Perimeter = 2L + 2W
Given that Mary has 150 ft of fencing, we have the equation:
2L + 2W = 150
Furthermore, we need to consider that Mary wants to leave a 10 ft opening on one side for a gate. This means that the length of the garden will be shorter by 10 ft:
L = W - 10
Now, we can substitute L into the first equation:
2(W - 10) + 2W = 150
Simplifying this equation:
2W - 20 + 2W = 150
4W - 20 = 150
4W = 170
W = 42.5
Now, we can substitute W back into the equation for L:
L = W - 10
L = 42.5 - 10
L = 32.5
So the dimensions of the garden that would result in the maximum area are approximately 32.5 ft by 42.5 ft.
2x+2y-10 = 150
x+y = 80
area = xy = x(80-x)
Just find the vertex of that parabola for maximum area.