Expert: Josh - 1/26/2005

Question
Problem:
The width of a rectangular playground is 75 percent of the length. If the perimeter of the playground is 280 meters, how long, in meters, is a straight path that cuts diagonally across the playground from one corner to another?

--Would you mind taking me through the steps? Thank you for your time--  

Answer
Kristi,

Here is the general guideline on how to formulate and solve a mathematical problem.

1. Always assign a variable (say, x, y etc.) to the unknown quality that you are trying to find.
2. Write down expressions to relate these different variables.
3. Solve the equations simultaneously to eliminate the unknowns down to a single variable.
4. Use back-substitution to find the remaining variables, once the value for one variable is determined.

Let us trace these steps and solve this problem.

Step 1: Instead of using "x", "y" etc., we'll use "w" to denote the width of the playground and use "l" to denote the length of the playground. So, "w" and "l" represent the unknown variables. Our aim is to find out the values of "w" (the width) and "l" (the length).

Step 2: relating "w" (the width) and "l" (the length) using the information given.
we are told that "w" (the width) is 75% of "l" (the length). Thus, w=0.75*l...call this equation [#1]

What else do we know.?
The perimeter (P) is clearly twice the sum of the width and length. So, P=2*(w+l)...call this equation [#2].
We are also given the value of P, perimeter P=280.
So, substitute this into [#2], and we get
280=2(w+l) ...call this [#2a].

Finally, using "d" to represent the length of the diagonal, from Pythagoras theorem, we know that
d= square_root_of(w^2+l^2) ... [#3]

Step 3: solving for either "w" or "l".
Now look at [#1] and [#2a]

From [#1], we have w=0.75*l,
From [#2a], we have P=280=2*(w+l),

just think about what we need to do...
what we started with is equation 1, which contains two unknowns. but with the second equation, we have two pieces of information and two unknowns. this allows us to eliminate "w" from equation 2.
plugging "w" into the second equation, only the variable "l" (the length) will remain in [#2a], thus, the value of "l" may be found.

280=2*(0.75*l+l)
  =2*(1.75*l)
  =3.5*l
Thus, l=280/3.5=80 meters.

Step 4: back-substitution.
Now, the value of "l" (the length) is known, we can find "w" simply from [#1].
w=0.75*80=60 meters.

Finally, work out the distance of the diagonal, "d".
d=sqrt(80^2+60^2)=sqrt(6400+3600)=sqrt(10000)=100.

What you need to learn from this is the general strategy and how to approach similar problems.

Cheers,
Josh