From Calculus, Varberg and Purcell, 6th Edition; am teaching myself from the text; no answer or explanation is given in the text.
I believe I was able to successfully obtain the correct (minimized) derivative for the value of a new variable I introduced ('x'- the distance from the base of the ladder to the fence) based on similar triangles, the height of the fence (simply 'h') and the fence's distance from the building ('w.')
Where I'm running into trouble is in applying my derived value of x in terms of general constants {x = (h^2*w)^(1/3)} back into the other equations (1 and 2) suggested by the diagram (i.e, finding the value of H, and then inserting that into the Pythagorean Theorem); for me the algebra is becoming intractable, and that's where I need help (if help is possible.) Here's the original question from the text:
"A fence h feet high runs parallel to a tall building, and w feet from the building. Find the length of the shortest ladder that will reach from the ground across the top of the fence to the wall of the building."
Note: I added three new variables, 'H', the height the ladder reaches on the building wall, 'x', the distance from the base of the ladder to the fence, and 'l', the length of the minimized ladder itself, my ultimate goal. My aim was to express all of these solely in terms of 'h' and 'w.' Note: there was a 'theta' symbol in the text's diagram implying a trigonometric approach, but I wanted to see if I could solve this algebraically.
Based on the diagram, similar triangles, and Pythagoras, I set up the following equations:
- l^2 =H^2 + (l+w)^2
- H/h = (x+w)/x H = h*(x+w)/x
Substituting (2) back into (1): l^2 = [h*(x+w)/x]^2 + (x =w)^2
Omitting a few of my steps, I simplified this to l^2 = (h + hw*x^-1)^2 + (x+w)^2
Next, the derivation, and setting the derivative equal to zero, to minimize:
2l dl/dx = 2 (h+hwx^-1) (-hwx^-2) + 2 (x+w) (1) = 0
dl/dx = -(h +hwx^-1)(hwx^-2) + (x+w) = 0
dl/dx = -(h^2wx^-2) - [(hw)^2*x^-3] + (x+w) = 0
h^2*w/x^2 (hw)^2 / x^3 = (x+w) / 1
h^2*w (x+w) /x^3 = (x+w) / 1 {the (x+w) term on each side should cancel}
h^2w /x^3 =1; h^2w = x^3; x = (h^2*w)^ (1/3), or x equals the cube root of h squared times w.
Note: I checked this partial solution with a similar online question giving numerical values for h (8) and w (4). their x was the cube root of 256, which is indeed the cube root of h squared times w. So I assumed I was in the right path.
However, I was now faced with substituting my value for the minimized x, expressed in terms of the given variables h and w, back into the original two equations, first H = h(x+w)/x, to obtain a value for H; and second, l^2 =H^2 + (l+w)^2 (Pythagoras), to obtain my ultimate quarry, the minimized value for 'l.'
Needless to say, the algebra, for me, was daunting. I'm sure there must be a more elegant trigonometric approach, as suggested by the diagram. I will spare any hopeful helpers any more algebra. I worked out an answer which I have no idea how to simplify further.
I believe the answer is in some sense 'correct,' in that if I carefully substitute in the numerical values for h (8), and w (4), I obtain a correct value (given in the online numerical problem) for l, which is approximately equal to 16.64775274.
However, my answer is unwieldy, to say the least, and if I were a math student (which I'm not), I'd be a bit scared to submit it. Here is my answer in terms of the given constants h and w; I have no idea how to take the square root, as calculators can't be used with general terms. If there is an algebraic approach, I probably stumbled off into the weeds early on. Any help would be appreciated. Here is my inelegant, unwieldy, and possibly useless answer:
the square root of {h^2 + 3[h^(4/3)*w^(2/3)+ h^(2/3)*w^(4/3)] + w^2}
There must be a simpler answer in terms of h and w, but it eludes me; I've lost my way. Once again, any help or advice is greatly appreciated.
