A ladder 10 feet long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall, and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to θ when θ=π/3?

Step-by-step explanation:

Given:

Length of ladder, y = 10 ft

θ = π/3 rad

Using cosine rule,

Sin θ = opposite/hypotenuse

Sin θ = x/y

Sin θ = x/10

Differentiating both sides with respect to dθ,

10 × cos θ × dθ = dx

10 cos θ = dx/dθ

dx/dθ = 10 × cos π/3

= 5 ft/radians.

answer:   14

step-by-step explanation:          

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horizontal translation of 2 units in the positive x direction

followed by

vertical translation of 3 units in the positive y direction