"Published in Newark, California, USA"
Each of two sides of a triangle are increasing at the rate of ½ foot per second, and the included angle is decreasing at 2° per second. Find the rate of change of the area when the sides and included angle are respectively 5 feet, 8 feet, and 60°.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
From Plane Geometry, the area of a triangle is given by the formula
From Trigonometry, we know that
Therefore,
Take the derivative on both sides of the equation with respect to time, we have
but b = 8 feet
c = 5 feet
θ = 60°
db/dt = dc/dt = ½ ft/sec
dθ/dt = - 2°/sec (negative because decreases)
= (- 2°/sec) x (π/180°) = - π/90 radians/sec
Substitute the values to the above equation, we have
Therefore,
Since the rate is positive, then the area is increasing.