SOLUTION: I've racked my brain for hours & I still don't know where to begin on this problem: A rectangular garden is to be surrounded by a walkway of constant width. The garden's dimensions

Question 90931This question is from textbook Beginning Al
: I've racked my brain for hours & I still don't know where to begin on this problem: A rectangular garden is to be surrounded by a walkway of constant width. The garden's dimensions are 30 ft by 40 ft. the total area, garden plus walkway, is to be 1800 ft^2. What must be the width of the walkway to the nearest thousandth?
Thank you. This question is from textbook Beginning Al

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
If we draw a picture we get

If we let x be the width of the path, notice there are 2 x-values per side. So we add 2x to 40 to get 40+2x. This is the total length of the side that contains 40 feet.

Also, this means we add 2x to 30 to get 30+2x. This is the total length of the side that contains 30 feet.

Start with the general area function:

Plug in (this is the total area)

Plug in and

Foil

Subtract 1800 from both sides

Let's use the quadratic formula to solve for x:

Starting with the general quadratic

the general solution using the quadratic equation is:

So lets solve ( notice , , and )

Plug in a=4, b=140, and c=-600

Square 140 to get 19600

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root (note: If you need help with simplifying the square root, check out this solver)

Multiply 2 and 4 to get 8

So now the expression breaks down into two parts

or

Now break up the fraction

or

Simplify

or

So these expressions approximate to

or

So our possible solutions are:
or

Since a negative length doesn't make sense, our only solution is

which is 3.860 to the nearest thousandth

Check:

Start with the given area function

plug in x=3.860

multiply

Add

Since we rounded, this is as close as it gets. So our answer is verified.

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