Finding the Rate of Change of Area in a Changing Triangle

[maymay43465]

Need help guys, not understanding this at all. Can anyone help me out?

Two sides of a triangle and their included angle are changing with respect to time. The angle increases at the rate of 1 radian/sec, one side increases at the rate of 3ft/sec, and the other side decreases at the rate of 2ft/sec. Find the rate at which the area of the triangle is changing when the angle is 45 degrees, the first is 4 feet long and the second side is 5 feet long. Is the area increasing or decreasing at this instant?

 

[HallsofIvy]

Since you are "given"the length of two sides and the angle between them, it shouldn't take too much to see that the cosine law gives you the length of the opposite side. Do you know a formula for the area of a triangle, given the lengths of the three sides? The derivative of both sides of that area formula, with respect to t should give you what you want.

 

[maymay43465]

question

Even if I am not sure I have a 90 degree angle this will work?

 

[HallsofIvy]

?? Did I say anything about a right angle? The "cosine law" works for any triangle, not just a right triangle. I was thinking about using the cosine law to find the length of the third side in terms of the other two and the angle between them, then using "Hero's formula" for the area.

Here is a much simpler way: first draw a picture. Draw your triangle with side of given length "a" as the base, the side with given length "b" going up from it and given angle \theta between them. You know, I hope, that the area of a triangle is "1/2 base times height". You already have "a" as the length of the base. The "height" is measured perpendicular to the base so drop a perpendicular from the end of the second side to the base. The second side is then the hypotenuse, of length b, of a right triangle with angle \theta. Use trig to write the "altitude", the length of that perpendicular as a function of b and \theta. Now write the equation for the area of the triangle as a function of a, b, and \theta. Differentiate that with respect to time to get the rate of change of area.