# Solution to a ladder problem. Calculus, minimizing the hypotenuse length.Basically, I am trying to solve problems from this...

Solution to a ladder problem. Calculus, minimizing the hypotenuse length.

Basically, I am trying to solve problems from this site:

http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxmindirectory/MaxMin.html#PROBLEM%2018

Using my own process, I came up to the answer of 14.99. However, the answer on the site is 17.99? How can that be? I noticed that in the last part of the solution, y=12.16, and it suddenly changed to 16.67? Thanks to anyone who can explain.

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### 1 Answer

You need to use the following notations such that: x represents the horizontal leg of right triangle formed by the ladder to x axis, y represents the vertical leg of right triangle formed by the ladder to y axis, L represents the length of the ladder and the hypotenuse of right triangle.

You need to draw a line,parallel to y axis, inside the right triangle, which has the length of 8.

The smaller right triangle, inside the larger one, has the vertical leg of 8, the horizontal leg of x - 3 and the length of hypotenuse of l.

You should notice that these triangles are similar, hence you may get the following relation between them legs such that:

`8/y = (x-3)/x`

`8x = y(x-3)`

`8x - xy = 3y =gt x(8 - y) = 3y`

`x = (3y)/(8-y)`

You need to use Pythagorean theorem to find the length of the ladder such that:

`L^2 = x^2 + y^2`

You need to substitute `(3y)/(8-y) ` for x such that:

`L^2 = ((3y)/(8-y))^2 + y^2`

`L^2 = (9y^2)/((8-y)^2) + y^2`

You need to minimize the length of the ladder, hence you need to differentiate L with respect to y such that:

`L(y) = sqrt((9y^2)/((8-y)^2) + y^2)`

` L'(y) =(((9y^2)/((8-y)^2) + y^2)')/(2sqrt((9y^2)/((8-y)^2) + y^2))`

`L' (y) = ((18y(8-y)^2+ 18y^2(8-y))/((8-y)^4) + 2y)/(2sqrt((9y^2)/((8-y)^2) + y^2))`

You need to factor out 18y(8-y) such that:

`L' (y) = ((18y(8-y)(8 - y + y))/((8-y)^4) + 2y)/(2sqrt((9y^2)/((8-y)^2) + y^2))`

`L' (y) = ((144y)/((8-y)^3) + 2y)/(2sqrt((9y^2)/((8-y)^2) + y^2))`

`L' (y) = (2y(72 + (8-y)^3))/(2sqrt((9y^2)/((8-y)^2) + y^2))`

Simplify the expression by 2:

`L' (y) = (y(72 + (8-y)^3))/(sqrt((9y^2)/((8-y)^2) + y^2))`

You need to remember that L(y) is minimum if L'(y)=0, hence you need to solve for y the equation L'(y)=0 such that:

`y(72 + (8-y)^3)) = 0 =gt y =` 0

`72 + (8-y)^3 = 0 =gt (8-y)^3 = -72`

`8 - y = root(3)(-72)`

`8 - y = -2root(3)(9) =gt y = 8 + 2root(3)(9)`

`y = 12.160`

**Hence, the length of the ladder is minimized at y = 12.160.**