Word problem?

Question

A 15-foot ladder is leaning against a wall. If the distance from the top of the ladder to the ground is three feet more than the distance from the bottom of the ladder to the wall, then what is the distance from the top of the ladder to the ground? Please explain your answer. Thanks!

Answer 1

Let x be the distance from the bottom of the ladder to the wall.
Let y be the distance from the top of the ladder to the ground.

First you are told that y is 3 more than x:
y = 3 + x

Next, we assume the wall is at a right angle to the floor which is flat. So you have a right triangle. The hypotenuse is the length of the ladder (15 ft.). Thus by the Pythagorean theorem:
x² + y² = 15²

So substitute the first into the second and you get:
x² + (x + 3)² = 15²

Expand the square:
x² + x² + 6x + 9 = 225

Group everything on the left:
2x² + 6x - 216 = 0

Divide both sides by 2:
x² + 3x - 108 = 0

Now factor:
(x + 12)(x - 9) = 0

So either x = -12 or x = 9. Obviously we ignore the negative result (since we are talking about lengths). That leaves x = 9.

And y = x + 3 = 12

The top of the ladder is 12 feet from the ground.

By the way, if you look closely, you'll see this is a 3-4-5 triangle, just tripled to 9-12-15. If you saw this initially, you could just skip all the algebra. :-)

Answer 2

If you were to draw a figure of this, you would essentially end up with a right triangle. We can solve this problem using the Pythagorean Theorem (a² + b² = c²; where a and b are the legs of the right triangle (the "shorter" sides) and c is the hypotenuse (the longest side")).

Let x = distance from the bottom of the ladder to the wall
Let (x + 3) = distance from the top of the ladder to the ground

(x)² + (x + 3)² = (15)² --- Simplify...
x² + (x² + 6x + 9) = 225 --- Yet more simplifying...
2x² + 6x + 9 = 225 --- Subtract 225 from each side...
2x² + 6x - 216 = 0 --- Factor out the 2...
2(x² + 3x - 108) = 0 --- Factor the trinomial...
2(x + 12)(x - 9) = 0 --- Divide both sides by 2...
(x + 12)(x - 9) = 0 --- Either (x + 12) = 0 or (x - 9) = 0... solve each for x...
x + 12 = 0 --- Subtract 12 from each side...
x = -12
x - 9 = 0 --- Add 9 to both sides...
x = 9

Since the problem calls for a positive answer (it can't be negative), and we have x = {-12, 9}, x must equal 9. Since (x + 3) = distance from the top of the ladder to the ground, the answer must be 12 feet.

ANSWER: The distance from the top of the ladder to the ground is 12 feet.

CHECK:
x = 9
(9)² + (9 + 3)² = (15)²
81 + (12)² = 225
81 + 144 = 225
225 = 225 --- Our answer checks out...

Answer 3

Because of the pythagorean theorem, we can write a^2 + (a+3)^2 = 15^2. Then solve. a = 9. because the ladder is represented by (a+3), the top of the ladder is 12 feet above the ground.

Answer 4

15^2=x^2+(x+3)^2
225=x^2+x^2+6x+9
2x^2+6x-216=0
(2x-18)(x+12)=0 since x>0 to have physical meaning
2x-18=0
x=9' from bottom of ladder to wall
x+3=12' from top of ladder to ground

Answer 5

wall: x+3
bottom: x
x^2+(x+3)^2= 225
x^2+x^2+6x+9=225
2x^2+6x -216=0
2(x^2+3x-108)=0
2(x+12)(x-9)
x= -12 and 9

distance can't be negative so 9 is the answer
wall: 9+3=12
bottom: 9

Answer 6

where X is the length of side C
Side A (Wall) = 3+x
Side B (Ladder)= 15
Side C (Ground)= x
for it to be a triangle (which it is) the sum of any two sides must be greater than the third side. therefore...
A+B>C
B+C>A
C+A>B

(3+x)+15>x
15+x>(3+x)
x+(3+x)>15

18+x>x
12+x>x
2x>12

18>0
12>0
x>6

A=18
B=15
C=6