Math CentralQuandaries & Queries


Question from Shaun, a student:

Redwood and Glendale Parks are each surrounded by 100 yards of fencing. The area of Redwood Park is 25 square yards more than the area of Glendale Park. If both parks are rectangles, what is the measurement of each rectangle?


There is not enough information here to solve the problem. Do you know =
anything else?


Shaun wrote back

My dad couldn't figure it out either. This problem was extra credit from my teacher and that was all she put on the board. Thank you for trying.



It's not that neither your dad or I could figure it out, the difficulty is that there are many possibilities for the dimensions of the parks. Suppose Redwood Park is x yards by y yards and Glendale Park is p yards by q yards. Then you know

2x + 2y + 2p + 2q = 100 yards, and
xy = pq + 25 square yards.

You have 4 unknowns and only two equations so I wouldn't expect to be able to find a unique solution.

Suppose for example that p = q = 10 yards, then

x + y = 30 and xy = 125.

From the first equation y = 30 - x and hence, substituting into the second equation gives

x(30 - x) = 125 or x2 - 30x + 125 = 0.

This factors as

(x - 5)(x - 25) = 0

and thus

x = 5 yards or 25 yards.

Going back to the original question you can verify that the Redwood Park being 5 yards by 25 yards and the Glendale Park being 10 yards by 10 yards is a valid answer.

But what if the Redwood Park is p = 10 yards by q = 5 yards. A similar analysis yields dimensions of 32.707 and 2.293 yards as the dimensions of the Glenwood Park. Another valid answer.

There are many valid answers.


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