Hello, I was just wondering if someone can double check my answer on this word problem.
Thank you.
You have a rectangular garden that is 25ft by 11ft. You want to build a pathway around the garden. However, you only have enough gravel to build 576 ft.
So, how wide can your pathway around the garden be?
What I did was find the perimeter. P=72 ft
So, then I just divided 576 by 72 and I got 8.
My answer was that the pathway had to be 8 ft wide.
Is that correct?
For a path of width w the total area (path plus enclosed area) is (25+2w)(11+2*w) and the enclosed area is (25)(11). The difference is given as 576.
eqn = (25 + 2 w)*(11 + 2 w) - (25*11) == 576;
soln = Solve[{eqn, w > 0}, w][[1]]
{w → 6}
eqn /. soln
True
So the path is 6 ft wide.
Alternatively we could do it this way, look at diagram
2 (11 * x) + 2 (25 * x) + 4 x^2 = 576
giving
4 x^2 + 72 x - 576 = 0
We could simplify it to
x^2 + 18 x - 144 = 0
Solve either by hand or
Solve[x^2 + 18 x - 144 == 0, x]
{{x -> -24}, {x -> 6}}
and clearly x can’t be -24 so there you have the answer.
Paul.
