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# help

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A ladder is leaning against a building. The distance from the bottom of the ladder to the building is 20 ft shorter than the length of the ladder. How high up the side of the building is the top of the ladder if that distance is 10 ft less than the length of the ladder?

Feb 2, 2020

#1
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I can give you a hint: try expressing one of the lengths as a variable and try to find the other two lengths in term of your variable. Then what can you do?

Feb 2, 2020
edited by Impasta  Feb 2, 2020
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Solve for x:
x^2 = (x - 20)^2 + (x - 10)^2

Write the quadratic polynomial on the right hand side in standard form.
Expand out terms of the right hand side:
x^2 = 2 x^2 - 60 x + 500

Move everything to the left hand side.
Subtract 2 x^2 - 60 x + 500 from both sides:
-x^2 + 60 x - 500 = 0

Factor the left hand side.
The left hand side factors into a product with three terms:
-(x - 50) (x - 10) = 0

Multiply both sides by a constant to simplify the equation.
Multiply both sides by -1:
(x - 50) (x - 10) = 0

Find the roots of each term in the product separately.
Split into two equations:
x - 50 = 0 or x - 10 = 0

Look at the first equation: Solve for x.
Add 50 to both sides:
x = 50 or x - 10 = 0

Look at the second equation: Solve for x.
Add 10 to both sides:

x = 50        or             x = 10{Discard}

x  = 50 feet - length of the ladder.

50 - 10 = 40 feet - The height of the side of the building to the top of the ladder.

50 - 20 =30 feet - The distance from the foot of the ladder to the wall.

Feb 2, 2020
#3
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Right triangle    hypotenuse = l    (length of ladder)      base leg = l-20    height leg = l-10

Pythgorean theorem

(l-20)2 + (l-10)2 = l2       Expand, simplify to

l2-60l+500 = 0      Quadratic formula yields    l = 50 ft   or 10 (throw out)

The Height leg =  l-10 = 50 - 10 = 40 ft

Feb 2, 2020