Page 160 - How To Solve Word Problems In Calculus

3
sin θ b
=
3
cos θ a
b
3
tan θ =
a

1/3
b b
3
tan θ = =
a a 1/3

Note: The angle θ such that tan θ = b 1/3 /a 1/3 is actually the
angle corresponding to the pipe of minimum length that will
not fit around the corner. As θ → 0or θ → π/2,
L →∞.

2
2
2
2
sec θ = tan θ + 1 csc θ = cot θ + 1
b 2/3 a 2/3
= + 1 = + 1
a 2/3 b 2/3
a 2/3 + b 2/3 a 2/3 + b 2/3
= =
a 2/3 b 2/3
√ √
a 2/3 + b 2/3 a 2/3 + b 2/3
sec θ = csc θ =
a 1/3 b 1/3

L = a sec θ + b csc θ

√ √
a 2/3 + b 2/3 a 2/3 + b 2/3
= a + b
a 1/3 b 1/3
√ √
= a 2/3 a 2/3 + b 2/3 + b 2/3 a 2/3 + b 2/3
√
= (a 2/3 + b 2/3 ) a 2/3 + b 2/3

)
= (a 2/3 + b 2/3 3/2

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