How much higher is the ladder now?

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2 Answers
Apr 22, 2018

#" "#
The ladder moves #color(red)("8 feet"# higher.

Explanation:

#" "#
#color(green)("Step 1"#

Observe the sketch below:
enter image source here

#color(green)("Step 2"#

Triangle #color(blue)(ABC# is a right-triangle.

#color(red)(AC# is the Hypotenuse, as #color(red)(AC# is the side opposite to the right-angle at #color(red)(B#.

#color(blue)("AC = 50 feet"#.

Let #color(blue)("AB = d feet"#.

#color(blue)("BC = 40 feet"#.

Using the Pythagoras' Theorem

#(AC)^2 = (AB)^2+(BC)^2#

#50^2=d^2+40^2#

#2500=d^2+1600#

#d^2+1600=2500#

Subtract #color(red)(1600# from both sides of the equation.

#d^2+1600-color(red)(1600)=2500-color(red)(1600#

#d^2+cancel 1600 - color(red)(cancel 1600)=900#

#d^2=900#

Take Square Root on both sides

#sqrt(d^2) = sqrt(900#

#color(blue)(d = 30#

#color(red)( :. " AB = 30 feet"#

#color(green)("Step 3"#

We know that the ladder (AC) moves up #color(red)"x feet"# from its initial position.

Observe the sketch given below:
enter image source here

Focus on the right-triangle #color(red)(DBE#.

#"CE = x feet"#.

#"DB = ( 30 - 2x ) feet"#, as #color(blue)("AB = 30 feet"# from the previous step.

#BE =BC+CE= (40+x)# feet.

#color(green)("Step 4"#

Using Pythagoras' Theorem

#color(blue)((DE)^2=(BD)^2+(BE)^2#

#50^2=(30-2x)^2+(40+x)^2#

Note: #color(green)((a+-b)^2=a^2 +- 2ab+b^2)#

#50^2={30^2-120x+(2x)^2} + {40^2+80x+x^2}#

#2500={900-120x+4x^2}+{1600+80x+x^2}#

#2500=4x^2+x^2-120x+80x+1600+900#

#2500=5x^2-40x+2500#

Subtract #color(red)(2500# from both sides

#5x^2-40x+cancel 2500-color(red)(cancel 2500)=2500-color(red)(2500#

#5x^2-40x=0#

#5x(x-8)=0# ( 5x is the common factor)

#(5x)=0 or (x-8)=0#

If #5x=0, x=0# ( Ignore - NOT suitable )

#x-8=0#

Add #color(red)(8# to both sides.

#x-cancel 8+color(red)(cancel 8)=0+color(red)(8#

#x=8#

Hence, #CE=8#

Hence, the ladder moves #color(red)("8 feet"# higher.

Jun 10, 2018

#x=8#

Explanation:

The ladder makes a right triangle with the wall. # 30^2 + 40^2=50^2 quad# is a variation on everyone's favorite Pythagorean Triple. So the initial position of the base is 30 feet from the wall.

After the top moves up #x# it's at #40+x.# The base is #30-2x.#

#(30 - 2x)^2 + (40 + x)^2 = 50^2#

# 30^2 - 120 x + 4x^2 + 40^2 + 80 x + x^2 = 50^2#

# 5x^2 - 40 x = 0#

#5x(x - 8) = 0#

We can rule out #x=0# so

#x=8#

Check:

# (30-2(8))^2 = 14^2 = 196#

#(40+8)^2=48^2=2304#

#196+2304=2500=50^2 quad sqrt#