Question 798185
THE MENTAL MATH SOLUTION:
{{{highlight(1foot)}}} because (3,4,5) (with {{{3^2+4^2=5^2}}}) is the most famous Pythagorean triple.
(It is the only one I can remember and a favorite of math teachers).
A Pythagorean triple is a set of 3 natural numbers (positive integers) that could be the measures of the sides of a right triangle, because the squares of the first two numbers add up to the square of the third number.
 
A 5 meter pole leaning against a wall, with the bottom at 3 meters from the wall has its top 4 meters high on the wall.
{{{drawing(300,300,-1,5,-1,5,
grid(0),rectangle(0,0,0.2,0.2),
red(line(3,0,0,4))
)}}} {{{3^2+4^2=9+16=25=5^2}}}
A 5 meter pole leaning against a wall, with the bottom at 4 meters from the wall has its top 3 meters high on the wall.
{{{drawing(300,300,-1,5,-1,5,
grid(0),rectangle(0,0,0.2,0.2),
red(line(4,0,0,3))
)}}} It's a flipped version of the same right triangle.
 
THE INTENDED SOLUTION:
Of course it is harder, as you should expect from a math teacher.
We find that the pole reaches a height of {{{h= 4}}} meters up the wall when its bottom is 3 meters from the wall by using the Pythagorean theorem.
{{{3^2+h^2=5^2}}}-->{{{9+h^2=25}}}-->{{{h^2=25-9}}}-->{{{h^2=16}}}-->{{{h=4}}}
As the base of the right triangle is increased by a distance {{{x}}} (in meters), from {{{3}}} to {{{3+x}}}, the height is decreased by a distance {{{x}}} (in meters), from {{{4}}} to {{{4-x}}}.
According to the Pythagorean theorem:
{{{(3+x)^2+(4-x)^2=5^2}}}
{{{9+6x+x^2+16-8x+x^2=25}}}
{{{25-2x+2x^2=25}}}
{{{2x^2-2x=0}}}
{{{2x(x-1)=0}}}
The solution {{{2x=0}}}-->{{{x=0}}} means not moving the pole at all, and is not what the problem requires.
The other solution, {{{x-1=0}}}-->{{{highlight(x=1)}}} means moving the bottom of the pole 1 meter away from the wall, so it will be 4 meters from the wall, which will get the top to reach 3 meters up the wall.