SOLUTION: I am not sure what category this problem should be under, please help! Hazel has a screen door whose height is 4 feet more than its width. She wishes to stabilize the door by at

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Question 79830: I am not sure what category this problem should be under, please help!
Hazel has a screen door whose height is 4 feet more than its width. She wishes to stabilize the door by attaching a steel cable diagonally. If the cable measures √194/2 ft, what are the dimensions of the door?
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Your problem tells you that the height (H) of rectangular screen door is 4 feet more than
its width (W). In equation form this becomes:
.
H+=+W+%2B+4
.
The problem also tells you that the diagonal (D) of this rectangular door is:
.
D+=+sqrt%28194%29%2F2
.
and (calculator time) the square root of 194 is 13.92838828. Divide this by 2 and you
find that the diagonal of the door (in feet) is:
.
D+=+6.964194139
.
Now you can apply the Pythagorean theorem which says that for a right triangle (formed
by the height, the width, and the diagonal of the door), the sum of the squares of each
of the legs of the triangle equals the square of the diagonal. For this problem you can
write the equation that:
.
H%5E2+%2B+W%5E2+=+D%5E2
.
But from your first steps on this problem you know that H = W + 4, so you can substitute
W + 4 for H and the equation becomes:
.
%28W%2B4%29%5E2+%2B+W%5E2+=+D%5E2
.
Square the (W + 4) term on the left and the equation becomes:
.
%28W%5E2+%2B+8W+%2B+16%29+%2B+W%5E2+=+D%5E2
.
Combine the two W%5E2 terms on the left side and the equation becomes:
.
2W%5E2+%2B+8W+%2B+16+=+D%5E2
.
Next recall that earlier you found that the diagonal D was 6.964194139 feet. Square this
and you get that:
.
D%5E2+=+%286.964194139%29%5E2+=+48.5
.
In the quadratic equation substitute 48.5 for D%5E2 and the equation becomes:
.
2W%5E2+%2B+8W+%2B+16+=+48.5
.
Get this into standard quadratic form by subtracting 48.5 from both sides. The result is:
.
2W%5E2+%2B+8W+-+32.5+=+0
.
Although you do not need to do this next step, you can make the problem a little easier
by dividing both sides by 2 just to eliminate the 2 that multiplies the W%5E2 term.
When you do that division the quadratic equation becomes:
.
W%5E2+%2B+4W+-+16.25+=+0
.
Now you can apply the quadratic formula to solve this equation. The quadratic formula
says that for a quadratic equation of the form:
.
ax%5E2+%2B+bx+%2B+c+=+0
.
the two solutions will be:
.
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
.
By comparing your equation with the standard form you see that the corresponding factors
are: x = W, a = +1, b = +4, and c = -16.25
.
Substitute these values into the solution equation and you get:
.
W+=+%28-%28%2B4%29+%2B-+sqrt%28+4%5E2-4%2A1%2A%28-16.25%29+%29%29%2F%282%2A1%29+
.
This simplifies to:
.
W+=+%28-4+%2B-+sqrt%2816+%2B+65%29%29%2F2
.
The term in the radical is 81 and its square root is 9. This simplifies your equation
to:
.
W+=+%28-4+%2B-+9%29%2F2+
.
This means your possible solutions are:
.
W+=+%28-4-9%29%2F2+=+-13%2F2+=+-6.5 and W+=+%28-4%2B9%29%2F2=+5%2F2+=+2.5
.
You can ignore the first solution because it is negative and having a negative width
for a screen door doesn't make sense. The second solution means that the width of the
door is 2.5 feet (or 30 inches). A little narrow by today's standard construction
but a reasonable answer. The height is 4 feet more than that (from your work at the
start of the problem) and that means the door is 2.5 + 4 = 6.5 feet (78 inches) tall.
.
Hope this helps you to understand the problem.
.