Question 855640
This was question 855638, question 855639, and question 855640.
Were there really 3 people asking the same question, or someone asked the same question 2 or 3 times in rapid succession?
 
a) A 6 foot man is standing atop a 9 foot wall The distance along the hypotenuse from the top of the man's head to the ground is 17 feet. Kneeling children are lined up at the base of the wall. Each kneeling child takes up 1 foot of horizontal space.
{{{drawing(200,300,-9.5,2.5,-1.5,16.5,
triangle(-15,0,5,0,-4,-20),green(rectangle(0,0,-0.4,0.4)),
rectangle(0,0,1.5,9),locate(-4,7.5,green(17)),
locate(-1,7.5,green(15)),locate(-4.2,1.5,green(x)),
circle(0.1,14.7,0.3),line(0.1,14.3,1,9),
line(0.2,9,0.5,12),green(triangle(-8,0,0,0,0,15)),
arc(-5.5,0,1,1.5,180,360),arc(-6.5,0,1,1.5,180,360),arc(-7.5,0,1,1.5,180,360),
arc(-0.5,0,1,1.5,180,360),arc(-1.5,0,1,1.5,180,360),arc(-2.5,0,1,1.5,180,360),
arc(-3.5,0,1,1.5,180,360),arc(-4.5,0,1,1.5,180,360)
)}}} There is a green right triangle with leg measures {{{x}}} and {{{15}}} feet, and hypotenuse measuring {{{17}}} feet.
According to the Pythagorean theorem
{{{x^2+15^2=17^2}}}-->{{{x^2+225=289}}}-->{{{x^2=289-225}}}-->{{{x^2=64}}}--> {{{x=sqrt(64)}}}--> {{{highlight(x=8)}}}
The horizontal space along the bottom of the triangle is {{{highlight(x=8)}}} ,
so {{{highlight(8)}}} kneeling children (each one taking up 1 foot of horizontal space fit on that leg of the right triangle.
 
b)The man jumps off the original wall and 4 children are removed from the left side of the line. A new triangle is formed. What is the length of the hypotenuse of the new triangle measured from the top of the wall to the ground?
{{{drawing(200,300,-9.5,2.5,-1.5,16.5,
triangle(-15,0,5,0,-4,-20),green(rectangle(0,0,-0.4,0.4)),
rectangle(0,0,1.5,9),locate(-2,4.5,green(y)),
locate(-0.5,4.5,green(9)),locate(-2.2,1.5,green(4)),
circle(-7.9,5.7,0.3),line(-7.9,5.3,-8,0),
line(-7.8,3,-7.5,0),green(triangle(-4,0,0,0,0,9)),
arc(-0.5,0,1,1.5,180,360),arc(-1.5,0,1,1.5,180,360),arc(-2.5,0,1,1.5,180,360),
arc(-3.5,0,1,1.5,180,360)
)}}} With {{{8-4=4}}} children left, the horizontal leg of the new right triangle measures only {{{4feet}}} .
The vertical lefg measures {{{9 feet}}} , the height of the wall.
The Pythagorean realation for the length in feet is:
{{{y^2=4^2+9^2}}}-->{{{y^2=16+81}}}-->{{{y^2=97}}}-->{{{y=sqrt(97)=9.85}}} (rounded).
The length of the hypotenuse of the new triangle is approximately
{{{highlight(9.85feet=9feet10inches)}}} .
 
c) Each child on the ground (from the original triangle in part a) lies down head to toe, taking up 3 feet each. The 6 foot man is still standing on top of the wall, but the wall is shorter now. If the distance along the hypotenuse from the top of the man's head to the ground is exactly 26 feet, what is the new exact height of the wall? 
{{{drawing(500,300,-27,3,-4,14,
triangle(-35,0,5,0,-4,-20),green(rectangle(0,0,-0.5,0.5)),
rectangle(0,0,1.5,4),locate(-12,5,green(26)),
locate(-2,5.5,green(6+w)),locate(-11.5,1.1,green(24)),
circle(0.1,9.7,0.3),line(0.1,9.3,1,4),
line(0.2,4,0.5,7),green(triangle(-24,0,0,0,0,10)),
circle(-2.8,0.22,0.2),circle(-5.8,0.22,0.2),circle(-8.8,0.22,0.2),
circle(-11.8,0.22,0.2),circle(-14.8,0.22,0.2),circle(-17.8,0.22,0.2),
circle(-20.8,0.22,0.2),circle(-23.8,0.22,0.2),
line(-2.8,0.03,-0.02,0.03),line(-0.02,0.4,-0.02,0.03),
line(-5.8,0.03,-3.02,0.03),line(-3.02,0.4,-3.02,0.03),
line(-8.8,0.03,-6.02,0.03),line(-6.02,0.4,-6.02,0.03),
line(-11.8,0.03,-9.02,0.03),line(-9.02,0.4,-9.02,0.03),
line(-14.8,0.03,-12.02,0.03),line(-12.02,0.4,-12.02,0.03),
line(-17.8,0.03,-15.02,0.03),line(-15.02,0.4,-15.02,0.03),
line(-20.8,0.03,-18.02,0.03),line(-18.02,0.4,-18.02,0.03),
line(-23.8,0.03,-21.02,0.03),line(-21.02,0.4,-21.02,0.03)
)}}} The horizontal leg measures {{{8*3=24}}}ft and the hypotenuse measures {{{26}}}ft.
If {{{w}} is the new height of the wall (in feet), the vertical leg measures {{{6+w}}}ft.
According to the Pythagorean theorem
{{{(6+w)^2+24^2=26^2}}}-->{{{(6+w)^2+576=676}}}-->{{{(6+w)^2=674-576}}}-->{{{(6+w)^2=100}}}--> {{{6+w=sqrt(100)}}}-->{{{6+w=10}}}-->{{{w=10-6}}}-->{{{highlight(w=4)}}}
The new height of the wall is {{{highlight(4feet)}}} .