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AP Calculus BC Full Practice Exam 4 Section II Part A

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At time $$t$$, $$0\\le t\\le 8$$, the velocity of a particle moving along the $$x$$-axis is given by $$v(t)=2+3\\cos t-5\\sin(2t)$$. (a) Is the particle moving left or right at $$t=3$$ seconds? Explain your reasoning. (b) What is the average velocity of the particle from $$t=0$$ to $$t=8$$? (c) What is the average acceleration of the particle from $$t=0$$ to $$t=8$$? (d) Given that $$p(t)$$ is the position of the particle at time $$t$$ and $$p(0)=2$$, find $$p(4)$$.

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A tank is draining water at a rate proportional to the amount of water in the tank. Thus, $$\\frac{dA}{dt}=kA$$, where $$A$$ is the amount of water remaining at time $$t$$. Initially, there were $$120$$ gallons in the tank, and after $$8$$ hours there were $$90$$ gallons. (a) Write an expression for $$A(t)$$, the amount of water in the tank at any time $$t$$. (b) How much water will remain after $$4$$ hours? (c) What is the average amount of water in the tank during the first $$12$$ hours? (d) After how many hours will the amount of water in the tank be $$50$$ gallons?

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