Subject: Math-polynomial word problems
Name: Eve
Who are you: Parent (Middle)

I have 2 problems for you which I am trying to help my daughter solve but am myself inadequate. Please help soon.

1. A surveyor's map shows a plan for the rectangular rose garden whose area is
a2+25ab-350b2. Find an algebraic expression for the length and width. If a=20 ft and b=10 feet, find the actual dimensions of the garden.

2.A square is enclosed in a circle. The area of the square is (4r2-32r+64)sq cm and the area of the circle is 484r2 sq cm. Write a polynomial in factored form to represent the difference of the two areas.

How do I solve these? Thanks

 

Hi Eve.

  1. I think there is something odd about that first question. Let's start by factoring your expression for the area (call the area A):

    A = a2+25ab-350b2
    A = (a+35b)(a-10b)

    This is what is odd: any two numbers that give the same product as (a+35b) times (a-10b) will also give the area of the garden (as well of course as a+35b and a-10b themselves) and EACH of these pairs could form a length and a width for a garden of this size. For example, if the expression for the area came out equal to 12, then the length and width of the garden can still be almost anything (3 by 4, 2 by 6, 120 by 0.1 and so on):

    Here's another strange thing: If a=20 ft and b=10ft, then
    A = 202 + 25(20)(10) - 350(102)
    A = -29600

    Which is rather nonsensical for the area of a rose garden, because such areas can't be negative!

    Perhaps there is a typo in your expression. If not, fire the surveyor!

  2. Let the area of the square be S = 4r2 - 32r + 64.
    Let the area of the circle be C = 484r2.

    The difference of these areas is either S-C or C-S. But since the square is enclosed by the circle, the circle is larger, so it is C-S that gives us a positive difference, so we'll calculate that. C - S = (484r2) - (4r2 - 32r + 64)
    C - S = 480r2+ 32r - 64

    Now you can factor this to get your answer. It's easiest to factor if you take a common factor of 32 out of it first.

Stephen La Rocque>