A swimming pool problem
The owners of a rectangular swimming pool want to surround the pool with a crushed-stone border of uniform width. They have enough stone to cover 74 square meters. How wide should they make the border if the pool measures 10 meters by 25 meters?
Do I take the area of the larger rectangle minus the area of the smaller rectangle to get the area of the border. Like this:
(10 meter x 25 meter)-(74 square meters)=176 square meters
Not sure what to do please help.
Thank You :)
Your idea is correct. The area of the larger rectangle minus the area of the smaller tringle equals the area of the border.
But what are the dimensions of the large rectangle? 10m by 25m? No. Those are for the smaller rectangle.
We are looking for the uniform width of the border. We don't know it yet, so let us call it x meter wide.
Now we can get the dimensions of the larger rectangle.
width = 10 +x +x = (10 +2x) meters
length = 25 +x +x = (25 +2x) meters
(10 +2x)(25 +2x) -(10)(25) = 74
10*25 +10*2x +2x*25 +2x*2x -10*25 = 74
The 10*25 cancels out,
20x +50x +4x^2 = 74
4x^2 +70x -74 = 0
Divide both sides by 2,
2x^2 +35x -37 = 0 ------------***
You can use the Quadratic Formula to solve for x, but I think you are into factoring now, so we factor that,
(2x +37)(x -1) = 0
2x +37 = 0
2x = -37
x = -37/2 = -18.5 m ----reject this because there are no negative dimensions, for one.
x -1 = 0
x = 1 meter ------------answer.
width = 10 +2(1) = 12 m
length = 25 +2(1) = 27 m
(12)(27) -(10)(25) =? 74
324 - 250 =? 74
74 =? 74
Yes, so, OK.
too cold to swim
Well, good try but you can see that what you want is a width and the answer you give is in square meter (so it is an area). That gives you a hint that you should try something else.
Now, the owners have 74 m^2 of stone so this surface is only about stones. My point is, if you look at the figure we see two rectangles, one is the pool (blue) and one is the pool plus the stones (gray and blue. It contains the pool). So the area that the stones take A(st.) is just the gray one. So we can say that A(big rectangle) = A(pool) + A(st.). That is because we do not put stones in the pool but just around it. So the area A(st.) is not an area of a rectangle but just a border.
A(pool) = 10 * 25 =250 m^2.
A(st.) = 74 m^2
A(big rectangle) = (x+25+x) * (x+10+x) = (2x+25x)(2x+10) = 4x^2+70x+250
where x is the width (as in the figure)
But we know that : A(big rectangle) = A(pool) + A(st.) so :
4x^2+70x+250 = 250 + 74 = 324. Since we cannot isolate x in this equation we try the quadratic trick by putting the equation=0 :
then 4x^2+70x-74=0 and we get by factorization or the quadratic equation : x=1 or x=-18.5
Of course we reject x=-18.5 because a length can not be negative so x=1=width of the stone border. Done!
Now, just to make it easier to understant: We had that the area of the pool was 250 m^2. Now, if we increase both dimensions of the pool by 2 m (1 meter on each side of the pool) we get that the new dimensions are 12 and 27 meters so the area of our big rectangle is 12*27=324 meters. We know that not all of the area of the big rectangle is covered by stones but just what is outside of the pool so 324-250=74 which is the area (amount expressed in terms of area) of the stones that the owners can fill.
Hope it helps...
To conclude : an area is a way to describe the surface that an object fills so it is not always a basic form like a triangle, a square or a rectangle but it can be many things and one of the forms that it can take is described in this problem : a border. So the point is that if you want to describe the area of something try to find the amount of surface that thing fills.