A homeowner wishes to fence in three adjoining garden plots, as shown in the figure. If each plot is to be 80 ft^2 in area, and she has 88 feet of fencing material at hand, what dimension should each plot have?
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4 w + 6 L = 88
w L = 80
so
L = 80/w
4 w + 6 (80/w) = 88
w + 6 (20/w) = 22
w^2 - 22 w + 120 = 0
w = [ 22 +/- sqrt (484 - 480) ]/2
w = [ 22 +/- 2]/2
w = 12 or w = 10
if w = 12
L = 6 2/3
4 w + 6 L = 88 check
if w = 10
L = 8
4 w + 6 L = 88 check
Oops. My bad. Didn't set things up correctly.
a square has a area of 5.75cm squared
what is the length of the square
Well, I guess you could say this homeowner is trying to "plot" her way to a beautiful garden!
To figure out the dimensions of each plot, let's break it down. Each plot is 80 ft^2 in area, so we can say that the length times the width of each plot equals 80 ft^2.
Let's call the length of each plot "L" and the width "W". So, we have LW = 80.
Now let's think about the fencing material. The homeowner has 88 feet of fencing material total.
Each plot has three sides that need to be fenced. Since there are three plots, that means we need to consider 9 sides in total (3 sides per plot).
Therefore, the total length of the fencing material we need is 9L.
But here's the funny part - we know that we only have 88 feet of fencing material, so 9L = 88.
Now we can solve for L. Dividing both sides of the equation by 9 gives us L = 88/9.
So, the length of each plot should be approximately L = 9.78 feet.
But hold on, we still need to find the width! We can substitute L = 9.78 into LW = 80 to solve for W.
9.78W = 80
Dividing both sides of the equation by 9.78 gives us W = 8.18 feet.
So, each plot should have dimensions of approximately 9.78 ft by 8.18 ft. I hope this helps make the homeowner's gardening dreams come true!
To find the dimensions of each plot, we will need to determine the perimeter of each plot. Since all three plots are adjoining, they share two sides with each neighbor plot and one side with the open space.
Let's determine the total perimeter of the three adjoining plots.
The length of each plot can be represented by "L", and the width by "W".
Since each plot is a rectangle, the perimeter is given by the formula: P = 2L + 2W.
For the three adjoining plots, the total perimeter is:
Total Perimeter = 2L + 2W + W + W
We are given that the total available fencing material is 88 feet. We can set up the equation: Total Perimeter = 88.
Plugging in the values, we have:
2L + 2W + W + W = 88.
Simplifying the equation, we get:
2L + 4W = 88.
Now, we also know that each plot has an area of 80 square feet, so L * W = 80.
We can solve this system of equations to find the values of L and W.
First, let's solve the area equation for L:
L = 80/W.
Now substitute this value into the perimeter equation:
2(80/W) + 4W = 88.
Simplifying:
160/W + 4W = 88.
Multiplying through by W:
160 + 4W^2 = 88W.
Rearranging and simplifying:
4W^2 - 88W + 160 = 0.
Simplifying further:
W^2 - 22W + 40 = 0.
Using the quadratic formula, we find two potential values for W: W = 2 or W = 20.
If W = 2, then L = 80/2 = 40.
If W = 20, then L = 80/20 = 4.
However, we want all three adjoining plots to be the same size, and since 4 feet is an impractical width for a garden plot, we can conclude that each plot should be 40 feet long by 2 feet wide.
Looks like 3 plots, each of width x and height y. So,
4x+2y = 88
xy = 80
x(44-2x) = 80
2x^2-44x+80 = 0
x^2-22x+40 = 0
(x-20)(x-2) = 0
x = 20
So, each plot is 20 by 4