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Joanne has a garden that is 9 feet long by 6 feet wide. She wants to increase the dimensions of the garden by the same distance each way and have the area of the new garden be five times the area of the present garden. How many feet should she add to both the width and the length?
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What you are going to end up doing here is setting up and solving a quadratic equation.
You know that you have to increase both dimensions by the same number -- we'll call that x.
So you are going to have
(x+9)(x+5) = 270
This is because you have to multiply the new dimensions to get the new area. And the new area is 54*5=270.
When you multiply out the left side, you end up with
And that can become
x^2+15x-216 = 0
So then you just have to factor and that would get you
(x+24)(x-9) = 0
Which means that
x = -24 or x = 9
Of course, you're not going to subtract 24 feet from each side, so the answer is that you need to add 9 feet to each side.
By data the garden is rectangular, 9'X6' wth 9*6 =54 sq ft in area.
Let the distance increased by x in each directions, the resulting dimension is (x+9+x)' X (x+6+x) ' = (9+2x)(6+2x) sq ft algebaically. But this is equal to 5 times the original area = 5*9*6= 270 sq feet. Therefore the required equation of this quiz is :
(2x+9)(2x+6) =270, or
4x^2+30x -216 = 0 or by dividing by 2,
2x^2+15x-108 = 0, or
2x^2+24x-9x-108 = 0 or
2x(x+12)-9(x+12) = 0 or
(x+12)(2x-9) = 0 or
x+12 = 0 or 2x=-9.
x=-12 or x = 4.5
So x = 4.5 is the right answer.
Length both sides increased by is 4.5+9+4.5 = 18 '
Breadth both sides increased by 4.5 is 4.5+6+4.5 = 15'
So the new are = 18*15 =270 which is 5 times the original (9*6=54).
In a rectangular figure the area is equal to width multiplied by length of the rectangle. Expressed in the form of an equation
A = w*l
Where A = area ,
w = width = 6 feet, and
l = length = 9 feet
A = 6*9 = 54 feet^2
If both length and width of the rectangle are increased by a common distance x, the area of the enlarged triangle (A') is given by the following equation.
A' = (w + x)*(l + x) = (6 + x)*(9 + x) = 54 + 15x + x^2
If the increase in area of rectangle is 5 times then:
A' = 5*A = 5*54 = 270 feet^2
Therefor 54 + 15x + x^2 = 270
Therefor: x^2 + 15x - 216 = 0
Therefor: x^2 + 24x - 9x - 216 = 0
Therefor: (x + 24)*(x - 9) = 0
Therefor x = 9 or x = - 24.
We reject the negative value of x. Therefor x = 9.
Thus we will need to add 9 feet to each of width and length of the original rectangle to increase its area 5 times.
Thus the dimensions of expanded rectangle are:
Width = 6 + 9 = 15 feet
Length = 9 +9 = 18 feet
Area = 15*18 = 270 feet^2
The 9x6 garden will be 54 sq. feet in area. Joanne wants the garden to be 5 times that size (54x 5 = 270 square feet)
How do I figure the equal amount of land to add to each side of the garden ?
we have to decide what to add to each side to get the size of garden we want.
(6x) (9x) =270 (FOIL)
54 + 9x + 6x + xx = 270
9x + 6x + xx = 216
15x + xx = 216 (this is a far as we can go with the terms we have)
To solve for x, you must reorder the terms to equal 0
-216 + 15x + xx = 0
Factor the trinomial by taking out 24 and 9 ( what multiplies to get 216 and adds to get 15?)
(-24+ -1x) (9+ -1X) = 0
When you multiply two terms and they equal 0, then one of the terms is a zero
-24 + -1x = 0
-1x= 24 or x = -24
Now solve for the other x...
9 + -1x = 0
-1x = -9 or x=9
you have two possible answers: -24 and 9
If you add 9 feet to each side, the demension will be 15 x 19.
18x15 = 270
4.5 feet can be added in each direction if you want the origianal garden in the center
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