SOLUTION: The area of a triangular sheet of paper is 185 square inches. One side of the triangle is 7 inches more than three times the length of the altitude to that side. Find the length of

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Question 951622: The area of a triangular sheet of paper is 185 square inches. One side of the triangle is 7 inches more than three times the length of the altitude to that side. Find the length of that side and the altitude to the side.
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
b=side; h=altitude; A=Area=(1/2)b*h
A=(1/2)b*h
185 sq in=(1/2)(3h+7in)(h) Multiply each side by 2.
370+sq+in=3h%5E2%2B7h Subtract 370 sq in from each side.
0=3h%5E2%2B7h-370+sq+in
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ah%5E2%2Bbh%2Bc=0 (in our case 3h%5E2%2B7h%2B-370+=+0) has the following solutons:

h%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%287%29%5E2-4%2A3%2A-370=4489.

Discriminant d=4489 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-7%2B-sqrt%28+4489+%29%29%2F2%5Ca.

h%5B1%5D+=+%28-%287%29%2Bsqrt%28+4489+%29%29%2F2%5C3+=+10
h%5B2%5D+=+%28-%287%29-sqrt%28+4489+%29%29%2F2%5C3+=+-12.3333333333333

Quadratic expression 3h%5E2%2B7h%2B-370 can be factored:
3h%5E2%2B7h%2B-370+=+3%28h-10%29%2A%28h--12.3333333333333%29
Again, the answer is: 10, -12.3333333333333. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B7%2Ax%2B-370+%29

ANSWER 1: Length of the altitude is 10 inches
b=3h+7in=3(10in)+7in=37 in ANSWER 2: The length of the side is 37 inches.
CHECK:
Area=1/2bh=1/2(37 in)(10 in)
185 sq in=1/2(370 sq in)
185 sq in=185 sq in