Page 143 - How To Solve Word Problems In Calculus

The law of cosines is often needed when dealing with problems
involving triangles other than right triangles.

2
2
2
c = a + b − 2abcos θ a c
θ
b
Observe that if θ = π/2, the law of cosines reduces to the theo-
2
2
2
rem of Pythagoras: c = a + b . Other trigonometric identities
will be discussed as needed in the examples that follow.
Related Rates
EXAMPLE 1
Two sides of a triangle are 5 and 10 inches, respectively. The
angle between them is increasing at the rate of 5 per minute.
◦
How fast is the third side of the triangle growing when the
angle is 60 ?
◦

Solution

5 x

θ
10

Let θ represent the angle between the sides of length 5 and 10
and let x represent the length of the third side of the triangle.
In any calculus problem involving derivatives, all angles and
rates must be expressed in radian measure.

dθ π dx π
Given: = radians per minute Find: when θ =
dt 36 dt 3

180 ◦ π
π
◦ ◦
5 = = radians 60 = radians
36 36 3
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