SOLUTION: A ladder is leaning against a building so that the top is 8 feet above the ground. The length of the ladder is 2 feet less than twice the distance of the bottom of the ladder from

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Question 1192717: A ladder is leaning against a building so that the top is 8 feet above the ground. The length of the ladder is 2 feet less than twice the distance of the bottom of the ladder from the building. Find the length of the ladder.
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Drawing:

a = x = horizontal leg
b = 8 = vertical leg
c = 2x-2 = hypotenuse

Apply the Pythagorean Theorem
a%5E2+%2B+b%5E2+=+c%5E2

x%5E2+%2B+8%5E2+=+%282x-2%29%5E2

x%5E2%2B64+=+4x%5E2-8x%2B4

0+=+4x%5E2-8x%2B4-64+-+x%5E2

0+=+3x%5E2-8x-60

3x%5E2-8x-60+=+0

The last equation is in the form ax%5E2%2Bbx%2Bc+=+0 with
a = 3
b = -8
c = -60

Use the quadratic formula
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-8%29%2B-sqrt%28%28-8%29%5E2-4%283%29%28-60%29%29%29%2F%282%283%29%29

x+=+%288%2B-sqrt%28784%29%29%2F%286%29

x+=+%288%2B-++++28%29%2F%286%29

x+=+%288%2B28%29%2F%286%29 or x+=+%288-28%29%2F%286%29

x+=+%2836%29%2F%286%29 or x+=+%28-20%29%2F%286%29

x+=+6 or x+=+-10%2F3+=+-3.33 approximately
Ignore the negative result because negative side lengths aren't possible.

This x value then leads to
ladder length = 2x-2 = 2(6)-2 = 12-2 = 10

We have a 6-8-10 right triangle.

Answer:
The ladder is 10 feet long

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The other tutor showed a perfectly good formal algebraic solution.

For a student just learning how to solve problems using algebra, I would recommend learning how to solve the equation 3x%5E2-8x-60=0 by factoring instead of using the quadratic formula. Going to the effort of doing the factoring gives you more useful brain exercise than just plugging numbers into a formula.

And if formal algebra is not required, you can solve this problem in a very little time using logical reasoning.

The distances given in the problem are whole numbers, so the lengths of all the sides of the triangle are very probably whole numbers. Since one leg of the right triangle is 8, it is almost certain that the right triangle will have side lengths of 6, 8, and 10. And looking at those lengths we quickly see that they satisfy the conditions of the problem.

ANSWER: the length of the ladder is the hypotenuse, which is 10 feet.