Related rates with right triangles

Warm-up

Three quick questions from earlier lessons.

Discover

A 5-m ladder leans against a wall; its foot is $a$ m from the wall. Slide the foot outwards at a steady pace.

A 5-m ladder from (a, 0) on the ground to the wall at x = 0. As a grows by equal steps, the top drops slowly at first, then faster and faster.

The foot moves at a steady speed. Does the top slide down at a steady speed?

No: it speeds up near the ground
Yes, the same speed as the foot

Key idea

Right triangles link distances by Pythagoras.

If

x^2 + y^2 = D^2

, then

x\,x' + y\,y' = D\,D'

(all with respect to

t

). A constant length (a ladder) has rate 0; a decreasing length has a negative rate.

Worked example

A 10-ft ladder: foot 6 ft from the wall, sliding out at 2 ft/s. How fast is the top moving?

y = 8

At that moment y = √(100 − 36).

2x\,x' + 2y\,y' = 0

Differentiate x² + y² = 100.

y' = -\tfrac{3}{2}

ft/s: the top slides down

6 · 2 + 8 · y' = 0.

Your turn, guided

Your turn: same ladder, when the foot is 8 ft out (so $y = 6$ ).

y' = -\tfrac{8}{3}

ft/s

8 · 2 + 6 · y' = 0.

Practice ★

A 15-ft ladder leans against a wall. Its foot slides away from the wall at 4 ft/s. How fast is the top moving when the foot is 9 ft from the wall? (ft/s; negative means down)
A 10-ft ladder leans against a wall. Its foot slides away from the wall at 3 ft/s. How fast is the top moving when the foot is 6 ft from the wall? (ft/s; negative means down)
A 15-ft ladder leans against a wall. Its foot slides away from the wall at 2 ft/s. How fast is the top moving when the foot is 9 ft from the wall? (ft/s; negative means down)

Watch out!

Practice ★★

Two cars leave the same crossing at noon: one drives north at 50 mph, the other east at 51 mph. How fast is the distance between them growing? (mph, exact)

A plane flies at an altitude of 6 km, horizontally away from a radar station, at 11 km/min. How fast is its distance to the station increasing when the horizontal distance is 8 km? (km/min)
A plane flies at an altitude of 9 km, horizontally away from a radar station, at 10 km/min. How fast is its distance to the station increasing when the horizontal distance is 12 km? (km/min)

Boss problem. A 15-ft ladder slides: its foot moves away from the wall at 1 ft/s. Look at the moment the foot is 9 ft from the wall.

Exit ticket

Three questions to check you have got it.

A 15-ft ladder leans against a wall. Its foot slides away from the wall at 4 ft/s. How fast is the top moving when the foot is 9 ft from the wall? (ft/s; negative means down)
A 10-ft ladder leans against a wall. Its foot slides away from the wall at 3 ft/s. How fast is the top moving when the foot is 6 ft from the wall? (ft/s; negative means down)
A 15-ft ladder leans against a wall. Its foot slides away from the wall at 2 ft/s. How fast is the top moving when the foot is 9 ft from the wall? (ft/s; negative means down)

Summary card

Right-triangle rates

$x^2 + y^2 = D^2 \Rightarrow x\,x' + y\,y' = D\,D'$ .
Find the missing side at that moment first.

Watch out: rates at right angles do not add; decreasing lengths have negative rates.

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