A ladder is leaning against a building so that the distance from the ground to the top of the ladder is 2 feet less than the length of the ladder. Find the leng

Question

A ladder is leaning against a building so that the distance from the ground to the top of the ladder is 2 feet less than the length of the ladder. Find the length of the ladder if the distance from the bottom of the ladder to the building is 6

feet.

Answer 1

Hi Emily from Price, UT...hope you had a great week in school.

Great word problem...every Algebra/Geometry student's favorite! Right?  NOT!

Anyway...I'll help you set it up and you can do the rest. I won't answer it for you, but help you understand how to do them, then when you do, you will be a pro. One good way to begin is to draw a picture of the image the word problem is describing, which will help in understanding what the problem is asking you to solve.

Okay...you have a ladder leaning against a wall. We are presuming the ground and the wall are at a right angle...so this tell us WHAT???

With a right angle, 2 lengths/arms, we have a triangle, with the ladder forming the hypotenuse...which tells us [another WHAT?] Draw the figure.

    from the ground is 6 feet...which is one arm of the right angle

    the ladder[which you're asked to solve] is a certain length  we'll say  'x'

    the other arm, the wall, is 2 feet less than the ladder, so  (x -2)

So...we have all the labels of the sides(arms) of the triangle formed by the ground, wall and ladder. So how would we solve for an unknown length of a right triangle. What was that formula...was it the Pythagorean Theorem?

Remember that???    a²  +  b²  =  c²   

     so now we can make some substitutions:

           a = 6

           b = (x - 2)

           c = x                  now let's make the substitutions...

             6²  + (x - 2)² = x²      then we complete the calculations

I hope you see what we are doing. If you don't and you see these kinds of questions again, you won't know how to solve them. Make sure you understand what is being done. Just giving you the answer does not help you. What if this comes up in a test?  I'll continue, but MAKE SURE you know why these things are being done!!  Complete the calculations:

                 6² + (x - 2)² = x²          What is next?

                 36  + (x - 2)(x -2)  =  x²    

 Multiply (x - 2) we use FOIL, correct! First, Outer, Inner, Last which gives you...What? I'll help... x² - 2x - 2x - 4   Right? Make sure you understand!

We then complete the operations   - 2x - 2x = - 4x    so then

                   x² - 4x + 4    we put it back into our equation...

                  36 + x² - 4x + 4  =  x²     

 rearrange the equation, combining like terms...

                 36 + 4 + x² - 4x = x²   complete the operation and like terms

                40 + x² - 4x = x²          continue with combining like terms

               40 + x² - x² - 4x = x²  - x²         complete the operation  

                40 + 0 - 4x =  0      rewrite equations and combine like terms

                     40 - 4x + 4x = 0 + 4x        rewrite equation

                      40  =  4x 

Now, you should be able to do the rest. Make sure you understand how to set up the problem. If you don't, you won't know how to set and solve these kinds of problems if they appear on a test. I can tell you similar ones will appear on a ACT or SAT test.

Also, you should know how to check you answer, what would you do to check you solution to see if its the correct answer.

If you need any further assistance contact me on Wyzant. I can help you with exercises.

 

 

Answer 2

1. Let x equal the length of the ladder.
2. Then the distance from the ground to the top of the ladder is (x-2).
3. The distance from the ladder to the building is 6.

 

The ground and the building wall form a right angle, therefore use Pythagorean Theorem.

 

• x^2 = (x-2)^2 +6^2
• x^2 = x^2 - 4x + 4 + 36
• x^2 = x^2 - 4x + 40

Subtract x^2 from both sides leaving.

• 0 = -4x + 40

Add -40 to both sides.

• -40 = -4x

Divide both sides by -4.

• -40/-4 = -4x/-4
•       10 FEET = x  THE LENGTH OF THE LADDER.