Word problem

Question

a ladder is leaning up against a building. the distance from the bottom of the ladder to the building is 8ft shorter than the length of the ladder. how high up the side of the building is the top of the ladder if the distance is 4ft less than the length of the ladder?

Answer 1

First, draw a picture.

 

This is a right triangle, so the Pythagorean Theorem is useful:   c² = a² + b²

 

The hypotenuse is the length of the ladder.  Let x= ladder length

 

 "the distance from the bottom of the ladder to the building is 8ft shorter than the length of the ladder"  means

    a = x-8

 

  "how high up the side of the building is the top of the ladder if the distance is 4ft less than the length of the ladder"  means

    b = x-4

 

O.K., the math:

   c² = a² + b²

   x² = (x-8)² + (x-4)²              [replace with known formulas]

   x² = x² - 16x + 64    +   x² - 8x + 16          [expand using F-O-I-L]

   x² = 2x² - 24x + 80             [collect terms]

   0 = x² - 24x + 80            [subtract x² from both sides]

   0 = (x - 20)(x - 4)        [either factor or use quadratic formula]

 

This means that either x=20 or x=4.

The value x=4 creates negative (thus, not allowable) values for a and b.  So, x=20.

 

c=20

 

The problem asks, "how high up the side of the building is the top of the ladder...?", so b=20-4=16

 

The top of the ladder is 16 ft up the side of the building.

 

- - - -

Note for later:  "Pythagorean triples" are three numbers that satisfy the Pythagorean Theorem.  (3,4,5) is a very famous one and quite useful in textbook, quiz, and test problems because the calculations are easy.

   (12,16,20) is just (3,4,5) multiplied by 4

Answer 2

Monique, a² + b² = c², right?

 

(c - 8)² + (c - 4)² = c²

c² - 16c + 64 + c² - 8c + 16 = c²

c² - 24c + 80 = 0

(c - 20)(c - 4) = 0

 

c - 20 = 0

c = 20

20 - 4 = 16

 

c - 4 = 0

c = 4

4 - 4 = 0, so this answer is out.

 

The ladder is 16 feet up the wall.