Probem36+MaxS,+David+AC

once again, my scanner is not working so I'll do my best to draw one in paint.

So here we have a fence (8ft tall) that is parallel to a wall and 4 ft a way from its base The fence is the purple line, and the ladder is the green line and the wall is the tall blue line. For our objective equation I set up similar triangles.

y/4 = 8/x, y= 32/x

Now we have to decide are we minimizing or maximizing, we are minimizing

the equation we for L we are going to minimize is L = ( (4 +x)^2 + (32/x + 8)^2 ) ^ 1/2 L= (16 + 8x + x^2 +1024/x^2 +512/x +64 ) ^1/2 L= ( x^2 + 8x + 512/x +1024/x^2 + 80 ) ^1/2 Now we take the first derivative to find the critical values: L' = (1/2) ( 2x + 8 - 512/x² - 2048/x³) ^(-1/2) /(x²+8x+512/x + 1024/x²+80)This little addition will change your work below!! See my commment! 0=2x + 8 - 512/x² - 2048/x³ get rid of fracions, simplify and factor ... 2x⁴+8x³-512x-2048 = 0 x⁴+4x³-256x-1024=0 x³( x + 4) - 2(128x-521) 2(128x-521)=x³( x + 4)

Here's where i got stuck and turned to my calculator

I agree completely with your set-up...if you hadn't forgot about the exponent I would agree completely with your work. Here is a little hint: if you are minimizing a square root, you can simply minimize the inside expression. Since the square root function is one-to-one and increasing, the smaller the radicand (the inside), the smaller the root. I would simply call the inside expression something like Q, and then minimize that. Also, I ended up with a very unsatisfying number for x, and had to rely on my calculator as well. My answer for what you called x was very close to your value -- I got x = 6.727, in case you want to check. I could be wrong, but that gave me a value of L = 16.668. Actually, I let the angle that the ladder makes with the ground be called theta, and I maximized the value of theta.

Go to he solver function and for input plug in : 2x⁴+8x³-512x-2048 = 0

Then it will ask for a guess, I put in 8, press enter and the answer is x = 6.3496 m

therefore L = ( (4 +6.3496 )² + (32/6.3496 + 8 )² ) ^1/2

L = 16.64 meters