SOLUTION: I'm having trouble with this one! A rectangular garden has dimensions 3m by 4m. A path is built around the garden. The area of the garden and path is 6 times as great as the area o

Question 712636: I'm having trouble with this one! A rectangular garden has dimensions 3m by 4m. A path is built around the garden. The area of the garden and path is 6 times as great as the area of the garden. What is the width of the path?
Found 3 solutions by jim_thompson5910, josgarithmetic, nerdybill:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
The area of the garden is 3*4 = 12 square meters.

Multiply this by 6 to get 12*6 = 72. We do this because "The area of the garden and path is 6 times as great as the area of the garden"

So the area of the garden+path = 72 square meters

So if x = width of the path, then the width goes from 3 to 3+2x meters. This is because you're adding x to each direction on the width. Similarly, the length jumps from 4 to 4+2x meters

The area of the garden+path in algebraic form is

but this is 72 square meters which is what we found earlier, so

Solve for x

Now use the quadratic formula to solve for x

Plug in , ,

or

or

or

or

Toss out the negative solution since a negative width of the path makes no sense at all.

So the only reasonable answer is 5/2 = 2.5 meters

Therefore, in the end, the width of the path is 2.5 meters
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
Enough information about garden, path around, how they relate; and we assume that except for the corners, width of the path around garden is uniform.
In case you have trouble imagining some of this, try drawing a picture.

Area of garden: m^2.
Area of garden plus path: m^2
Area of garden plus path is 6 times area of garden alone:

That should be enough to understand the solution process for this example. w is the width of the path. Solve for w.
Answer by nerdybill(7384)   (Show Source): You can put this solution on YOUR website!
A rectangular garden has dimensions 3m by 4m. A path is built around the garden. The area of the garden and path is 6 times as great as the area of the garden. What is the width of the path?
.
Let x = width (m) of path
then
area of "path+garden" = (2x+3)(2x+4)
area of "garden" = 3*4 = 12
.
From:"The area of the garden and path is 6 times as great as the area of the garden." we get our equation:
(2x+3)(2x+4) = 6*12
4x^2+8x+6x+12 = 72
4x^2+14x+12 = 72
4x^2+14x-60 = 0
2x^2+7x-30 = 0
rewrite middle term:
2x^2+12x-5x-30 = 0
group:
(2x^2+12x)-(5x+30) = 0
2x(x+6)- 5(x+6) = 0
(x+6)(2x-5) = 0
x = {-6, 5/2}
we can throw out the negative solution (extraneous) leaving:
x = 5/2
or
x = 2.5 m

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