Posted by **Joe Schmo** on Monday, January 3, 2011 at 8:18pm.

A fence 8 ft. tall runs parallel to a tall building at a distance 4 ft. from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

- Calculus -
**drwls**, Monday, January 3, 2011 at 10:22pm
Let L be the ladder length and A be the angle that the ladder makes with the ground. Convince yourself that

L = 4/cosA + 8/sinA

dL/dA = (4/cos^2A)*sinA -(8/sin^2)*cosA

= 0 for minimum ladder length

sin/cos^2A = 2 cosA/sin^2A

sin^3A = 2 cos^3A

tanA = 2^1/3 = 1.260

A = 51.6 deg

Solve for L using the first formula above.

- Calculus -
**Joe Schmo**, Tuesday, January 4, 2011 at 7:38pm
Thank you very much! I just have one more question for you. How did you get L = 4/cosA + 8/sinA?

- Calculus -
**marg**, Wednesday, February 8, 2012 at 1:14am
a wall 10 feet high is 8ft from a house. what is the length of the shortest ladder that will reach the house when one end rests on the ground outside the wall?

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