A fence 3 feet tall runs parallel to a tall building at a di

Solution

First, draw a diagram. Let\'s call the ladder length \"x\", the height the ladder reaches up the wall \"h\", and the distance from the other (grounded) end of the ladder to the fence \"a\". By the Pythagorean Theorem we have: x^2 = (a + 5)^2 + h^2 (Hopefully you recognize the \"^\" symbol, which refers to an exponent, meaning something is raised to a power) Also, by similar triangles we can see: a/2 = (a + 5)/h Combining these two equations produces: x = sqrt(a^2 + 10a + 29 + 40/a +100/a^2) Now we need to use a bit of calculus and find the derivative of x with respect to a: dx/da = (2a + 10 + (-1)40/a^2 + (-2)100/a^3) / sqrt(a^2 + 10a + 29 + 40/a +100/a^2) Intuitively (in real world terms), this amounts to sliding the base of the ladder (\"a\" feet from the fence) around and seeing how fast the length of the ladder must change to reach the wall. When \"a\" is very large, the ladder doesn\'t have to reach high to get over the wall, but it has to be long because the fence is far away. Conversely, when \"a\" is very small, the ladder is closer to the wall (although still behind the fence), but it must go up at a very high angle and, consequently, reach high up on the wall. Either choice produces a long ladder. The shortest one is in between, when the rate of change is zero. Conveniently, setting the equation to zero means the numerator is zero, so denominator is irrelevant. This simplifies the math greatly. We can also multiply through by a^3 to make it prettier, since zero will remain zero on the left side of the equation: 0 = a^4 + 5a^3 - 20a - 100 Find the roots of the equations. I didn\'t have Matlab or my HP48 calculator handy, so I used excel, as described at: http://people.revoledu.com/kardi/tutoria… One of the roots is at -5, which is negative, so it\'s clearly not on we are looking for. The correct one is at 2.714. Plugging this into the original equation returns a ladder length x = 9.582 feet. So, in the end, we find that we should have just listened to the guy who spent 25 years painting houses, avoided all this hassle, and used a 10-foot ladder.