The problem:
A beam must be carried around a corner from a hall of width A to a hall of width B. What is the length of the longest beam that will turn the corner?
What I've done so far:
Let the length of the portion of the beam that extends into the hall of width A be P, let the length of the portion that extends into the hall of width B be Q, and let the angle that the beam forms with the inside wall of the hall of width B as it turns the corner be ?. Let L be the beam length (L = P + Q).
sin ? = B/Q ? Q = B/(sin ?)
sin (?/2 – ?) = cos ? = A/P ? P = A/(cos ?)
L = A sec ? + B csc ?
d/d? (A sec ? + B csc ?) = A sec ? tan ? – B csc ? cot ? ? 0
? A ? (sin ?)/(cos[sup:3pnwojev]2[/sup:3pnwojev]?) = B ? (cos ?)/(sin[sup:3pnwojev]2[/sup:3pnwojev]?)
? (sin[sup:3pnwojev]3[/sup:3pnwojev]?)/(cos[sup:3pnwojev]3[/sup:3pnwojev]?) = tan[sup:3pnwojev]3[/sup:3pnwojev]? = B/A
? tan ? = B[sup:3pnwojev]1/3[/sup:3pnwojev]/A[sup:3pnwojev]1/3[/sup:3pnwojev] = (B/Q)/(A/P) = (BP)/(AQ)
? B[sup:3pnwojev]1/3[/sup:3pnwojev]AQ = A[sup:3pnwojev]1/3[/sup:3pnwojev]BP
? A[sup:3pnwojev]2/3[/sup:3pnwojev]Q = B[sup:3pnwojev]2/3[/sup:3pnwojev]P
? Q = (B/A)[sup:3pnwojev]2/3[/sup:3pnwojev]P
L = P + Q = P + (B/A)[sup:3pnwojev]2/3[/sup:3pnwojev]P = P[1 + (B/A)[sup:3pnwojev]2/3[/sup:3pnwojev]]
My question:
I'm stuck here and can't see how to solve for L in terms of A and B. Any hints?
A beam must be carried around a corner from a hall of width A to a hall of width B. What is the length of the longest beam that will turn the corner?
What I've done so far:
Let the length of the portion of the beam that extends into the hall of width A be P, let the length of the portion that extends into the hall of width B be Q, and let the angle that the beam forms with the inside wall of the hall of width B as it turns the corner be ?. Let L be the beam length (L = P + Q).
sin ? = B/Q ? Q = B/(sin ?)
sin (?/2 – ?) = cos ? = A/P ? P = A/(cos ?)
L = A sec ? + B csc ?
d/d? (A sec ? + B csc ?) = A sec ? tan ? – B csc ? cot ? ? 0
? A ? (sin ?)/(cos[sup:3pnwojev]2[/sup:3pnwojev]?) = B ? (cos ?)/(sin[sup:3pnwojev]2[/sup:3pnwojev]?)
? (sin[sup:3pnwojev]3[/sup:3pnwojev]?)/(cos[sup:3pnwojev]3[/sup:3pnwojev]?) = tan[sup:3pnwojev]3[/sup:3pnwojev]? = B/A
? tan ? = B[sup:3pnwojev]1/3[/sup:3pnwojev]/A[sup:3pnwojev]1/3[/sup:3pnwojev] = (B/Q)/(A/P) = (BP)/(AQ)
? B[sup:3pnwojev]1/3[/sup:3pnwojev]AQ = A[sup:3pnwojev]1/3[/sup:3pnwojev]BP
? A[sup:3pnwojev]2/3[/sup:3pnwojev]Q = B[sup:3pnwojev]2/3[/sup:3pnwojev]P
? Q = (B/A)[sup:3pnwojev]2/3[/sup:3pnwojev]P
L = P + Q = P + (B/A)[sup:3pnwojev]2/3[/sup:3pnwojev]P = P[1 + (B/A)[sup:3pnwojev]2/3[/sup:3pnwojev]]
My question:
I'm stuck here and can't see how to solve for L in terms of A and B. Any hints?