4. Two sides of a triangle have lengths 12 m and 14 m. The angle between them is

Question

4. Two sides of a triangle have lengths 12 m and 14 m. The angle between them is increasing at a rate of 2 degree /min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60 degree? (Round your answer to three decimal places.) m/min 5. Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 700 cm3, the pressure is 140 kPa, and the pressure is increasing at a rate of 10 kPa/min. At what rate is the volume decreasing at this instant? Cm^3/min 6. Two people start from the same point. One walks east at 3 mi/h and the other walks northeast at 4 mi/h. How fast is the distance between the people changing after 15 minutes? (Round your answer to three decimal places.) mi/h

Explanation / Answer

1)

Law of Cosine: WE HAVE SIDE-ANGLE-SIDE.

c^2 = a^2 + b^2 - 2abCos C

The sides a and b are fixed, a = 12 and b = 14. c is the changing side length and C is the angle opposite of it.

c^2 = 12^2 + 14^2 - 2(12)(14)Cos C

c^2 - 144 + 196 - 336*Cos C

d/dt[c^2 = 340 - 336*Cos C]

2c dc/dt = -336*(-Sin C) dC/dt <===Derived related rate equation.

** Given: dC/dt = 2 and C = 60

Also if C = 60, then c^2 = 340

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