Area Of Walkway Around Rectangular Swimming Pool

G

Goddyhead

New member
Joined
Nov 26, 2017
Messages
1
Problem: "A 35- by 16-ft rectangular swimming pool is surrounded by a walkway of uniform width. If the total area of the walkway is 280 ft^2, how wide is the walkway?"

Now, I understand most of this, up until a snag at the very end. Here is the work I did so far:

I know you subtract the measurements of the swimming pool from the measurements of the outer rectangle, containing both the swimming pool and the walkway surrounding it.

The length of the outer rectangle is given by the expression (2x + 35), and its width is given by the expression (2x + 16). The area of the swimming pool is 560 ft^2, so we get the following equation:

(2x + 35)(2x + 16) - 560 = 280

FOILing, I was able to get it to…

4x^2 + 32x + 70x + 560 - 560 = 280
Simplified to…
4x^2 + 102x = 280
Subtracting 280 from both sides, I got…
4x^2 + 102x - 280 = 0
Dividing everything on both sides by 2, I got…
2x^2 + 51x - 140 = 0

Now here's the kicker. The next step is to factor this quadratic equation. However, there is a little issue that confuses me. Obviously, 2x^2 can be factored as being the product of 2x and x, but -140 has so many different factors that I don't know which two I am supposed to choose when factoring. I did 14 and -10, with the result that the quadratic equation was factored into the form (x + 14)(2x - 10) = 0. I then proceeded to set each term in each set of parentheses equal to 0 and solve for x, but the computer program said it was incorrect. When I gave up, unable to reason why my answer was incorrect, and asked it to show me the correct answer, it turned out that they had used different factors of -140 besides -10 and 14.
I suppose, if I had unlimited tries on the problem, I could go through all of the factors of -140 and plug them all in, and see if the computer program accepts them, but I cannot do that, as I can only get the answer wrong twice before I lose all points for that question.

So does anyone know how, in a problem of this nature, I am supposed to determine which factors of a number I am supposed to use and put in parentheses when factoring a quadratic equation, in a case like this, when the number has so many different factors to choose from?

Thank you very much.
 
Goddyhead said:
(2x + 35)(2x + 16) - 560 = 280

FOILing, I was able to get it to…
Click to expand...

That's good. You got ALL the pieces. "FOIL" is not a verb. Just learn to multiply.

Dividing everything on both sides by 2, I got…
2x^2 + 51x - 140 = 0
Click to expand...

Perfect.

Now here's the kicker. The next step is to factor this quadratic equation.
Click to expand...

Why do you think it can be factored? The idea is to SOLVE the equation. You shouldn't care exactly HOW you do that. Read my signature, below.


So does anyone know how, in a problem of this nature, I am supposed to determine which factors of a number I am supposed to use and put in parentheses when factoring a quadratic equation, in a case like this, when the number has so many different factors to choose from?
Click to expand...

This is why we invented the Quadratic Formula. Heard of it?

BTW: "Uniform Width" Does anything SAY the Pool+Walkway is RECTANGULAR?
 
Goddyhead said:
...
2x^2 + 51x - 140 = 0

Now here's the kicker. The next step is to factor this quadratic equation. However, there is a little issue that confuses me. Obviously, 2x^2 can be factored as being the product of 2x and x, but -140 has so many different factors that I don't know which two I am supposed to choose when factoring. I did 14 and -10, with the result that the quadratic equation was factored into the form (x + 14)(2x - 10) = 0. I then proceeded to set each term in each set of parentheses equal to 0 and solve for x, but the computer program said it was incorrect. When I gave up, unable to reason why my answer was incorrect, and asked it to show me the correct answer, it turned out that they had used different factors of -140 besides -10 and 14.
I suppose, if I had unlimited tries on the problem, I could go through all of the factors of -140 and plug them all in, and see if the computer program accepts them, but I cannot do that, as I can only get the answer wrong twice before I lose all points for that question.

So does anyone know how, in a problem of this nature, I am supposed to determine which factors of a number I am supposed to use and put in parentheses when factoring a quadratic equation, in a case like this, when the number has so many different factors to choose from?

Thank you very much.
Click to expand...

Evidently the only way you are familiar with for solving a quadratic equation is to factor, and the only way you know to factor a trinomial is the trial and error method. When you "tried" (x + 14)(2x - 10), you failed to expand this to check whether it is correct; it is equal to 2x^2 + 18x - 140, not to 2x^2 + 51x - 140. So this was an error - you don't just try one possibility and move on. And it is an error to try solving an equation using an unchecked (incorrect) factorization. It is also an error to let the homework program do all the checking for you. That's part of your job.

The complete trial and error method involves more things to try; it is indeed harder with big numbers like these. Another method that I prefer when things get complicated is called the "ac method", or "ac/grouping". You can search for that term; here is an explanation I wrote of why it works, including an example of how to do it: Factorization by Decomposition and the Distributive. This method still involves big numbers, but it is easier to organize your thinking. For a description of the trial and error method, together with a fuller explanation of the ac method, try this, or this. If you search enough, you can find good explanations of the complete trial and error method, complete with hints on how to do it with big numbers. But I wouldn't use the method for this problem anyway.

And as tkhunny pointed out, when it's really bad, the quadratic formula is usually the way to go.

What methods have you been taught? They wouldn't spring a problem like this on you without giving you the tools you need!
 
tkhunny said:
Nope. It says the POOL is rectangular, not the Pool+Walkway. Try again.
Click to expand...
If a walkway of uniform width is placed around a rectangular shape, how could the larger shape be non-rectangular? :???:
 
Indeed. But it doesn't SAY that. Thus, my question.
 
You must log in or register to reply here.
Top