Distance

[Guest]

A fence encloses a rectangular building whose perimeter (building) is 140 feet. The distance of the fence from the building is 1/3 the width of the building. The area between the fence and building is 1800 sq. ft. Determine distance the fence is from the building.

How do I proceed ? Thanks.

[Guest]

There are essentially 3 variables in this question:
W = the width of the building
L = the length of the building
x = the distance of the fence to the building
Our aim is to find x.

The question gives you various bits of information that can be rewritten as formulas involving these variables.

"building whose perimeter (building) is 140 feet" translates to:
2W+2L = 140

"The distance of the fence from the building is 1/3 the width of the building" translates to:
x = W/3

"The area between the fence and building is 1800 sq. ft" this one is harder to work out.
The building has dimensions W and L and so covers an area WL.
The fence has dimensions W+2x and L+2x (it is x feet from either wall of the building). So the fence covers an area of (W+2x)(L+2x).
The area between the fence and the building is the difference in the areas they cover, so:
(W+2x)(L+2x) - WL = 1800

So we have 3 formulas from the question, the second tells us that W = 3x, so we can substitute this into the other two equations to get:
6x+2L =140
5x(L+2x) - 3xL = 1800
The first rearranges to tell us that L = 70-3x. Substituting this into the third equation gives:
5x(70-x)-3x(70-3x) = 1800
This rearranges to give the following quadratic equation in x
[tex]4x^2+140x-1800 = 0[/tex]
This can be easily factorized and you should find the (sensible) solution is x = 10 ft.

Hope this helped,

R. Baber.

[Guest]

"6x+2L =140
5x(L+2x) - 3xL = 1800
The first rearranges to tell us that L = 70-3x. Substituting this into the third equation gives:
5x(70-x)-3x(70-3x) = 1800"

I don't understand these steps. Thanks.

[Guest]

We know W=3x and 2W+2L=140. The first tells us that we can replace every occurrence of W with 3x since they are the same. In particular we can replace the W in the equation 2W+2L=140 so that it becomes 2(3x)+2L=140, removing the brackets gives 6x+2L=140.

Similarly we can replace the W in the equation (W+2x)(L+2x) - WL = 1800 so that it becomes
((3x)+2x)(L+2x) - (3x)L = 1800
removing the brackets gives
(3x+2x)(L+2x) - 3xL = 1800
3x+2x is the same as 5x so this simplifies to
5x(L+2x)-3xL = 1800

We've shown previously that 6x+2L=140 we can use this to write L in terms of x.
The aim is that we want only the term L on one side, and the rules of the game is we must do the same thing on both sides of the equal sign to keep the equation balanced.
First subtract 6x from both sides to get
6x+2L-6x = 140-6x
On the left hand side the 6x terms cancel leaving
2L = 140-6x
Now divide both sides by 2
2L/2 = (140-6x)/2
This simplifies to
L = (140-6x)/2
We can simplify the right hand side by expanding the bracket
L = 140/2-6x/2
which becomes
L = 70-3x

Because L=70-3x we can as before replace occurrences with L with 70-3x. If we do this to 5x(L+2x)-3xL = 1800 we get
5x((70-3x)+2x)-3x(70-3x) = 1800
expanding the left bracket gives
5x(70-3x+2x)-3x(70-3x) = 1800
which is the same as
5x(70-x)-3x(70-3x) = 1800

Hope this helped,

R. Baber.

[Guest]

That helped. I understand now. Thanks again.

[Guest]

Just out of interest it might surprise you to know that we don't actually need the information that "The distance of the fence from the building is 1/3 the width of the building".

The area between the fence and the building gives:
[tex](W+2x)(L+2x) - WL = 1800[/tex]
which simplifies to
[tex]4x^2 + (2W+2L)x - 1800 = 0[/tex]
and we know the perimeter [tex]2W+2L[/tex] is 140, so we can substitute that in to get:
[tex]4x^2 + 140x - 1800=0[/tex]
and we haven't made any use of the fact that [tex]x=W/3[/tex].

Reminds me a lot of the area of an Annulus formula where the area doesn't depend on the radius of circles but the distance from the inner circle to the outer circle along a tangent ([tex]A = \pi d^2[/tex] see https://en.wikipedia.org/wiki/Annulus_%28mathematics%29 ).

R. Baber.

[Guest]

Thanks for the additional info.

"The area between the fence and the building gives:
(W+2x)(L+2x)−WL=1800
which simplifies to
4x 2 +(2W+2L)x−1800=0
and we know the perimeter 2W+2L is 140, so we can substitute that in to get:
4x 2 +140x−1800=0"

A question:

Did you use the same procedure for these steps as before ?

[Guest]

Yes.

R. Baber.

[Guest]

Thanks again.