# Calculus

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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 -

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 -

Thank you very much! I just have one more question for you. How did you get L = 4/cosA + 8/sinA?

• Calculus -

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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