SOLUTION: 1. A rectangular garden has dimensions of 18 feet by 13 feet. A gravel path of uniform width is to be built around the garden. How wide can the path be if there is enough gravel fo
Question 155376: 1. A rectangular garden has dimensions of 18 feet by 13 feet. A gravel path of uniform width is to be built around the garden. How wide can the path be if there is enough gravel for 516 square feet? Answer by jim_thompson5910(35256) (Show Source):
First let's find the area of the garden only: Area of Garden = 18*13=234
So the area of the garden only is 234 square feet.
Now let's draw a picture:
From the picture, notice that the length of the entire rectangle (including the width of the path) is (notice there are two "x" lengths per side) and the total width is .
So the area of the entire enclosure (including the path) is the expression
Now subtract off the area of the garden (we only want the area of the path)
FOIL
Combine like terms
So the area of the path only is
Start with the area of the path
Plug in (which is the area of the path)
Subtract 516 from both sides
Notice we have a quadratic equation in the form of where , , and
Let's use the quadratic formula to solve for x
Start with the quadratic formula
Plug in , , and
Square to get .
Multiply to get
Rewrite as
Add to to get
Multiply and to get .
Take the square root of to get .
or Break up the expression.
or Combine like terms.
or Simplify.
So the answers are or
Since a negative width doesn't make sense, this means that the only solution is