Page 75 - How To Solve Word Problems In Calculus

2.

A

z y

B x P
dx dy dz
Given: = 40 = 30 Find: 2 hours later
dt dt dt
From the diagram,
2
2
z = x + y 2
dz dx dy
2z = 2x + 2y
dt dt dt

dz dx dy
z = x + y
dt dt dt
Two hours after the cars leave their initial point, x = 80 miles
(2 × 40 = 80) and y = 60 miles (2 × 30 = 60). It follows that
2
2
z = 80 + 60 2
= 6400 + 3600
= 10,000

z = 100
dx dy
Substituting x = 80, y = 60, z = 100, = 40, and = 30, we
dt dt
obtain
dz
100 = 80 · 40 + 60 · 30
dt

dz
100 = 5000
dt
dz
= 50
dt

The distance between the cars is increasing at the rate of 50 mi/h.
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