Section II Part A — Full Practice Exam 2

Questions in this quiz (2)

1. The temperature on a winter day in Northfield is modeled by:$$T(H)=C+D\sin\left(\frac{\pi H}{24}\right)$$where $$T$$ is the temperature in degrees Fahrenheit and $$H$$ is the number of hours from midnight, $$0\le H\le24$$.a. The temperature at midnight is $$10$$ and the temperature at noon is $$30$$. Find $$C$$ and $$D$$.b. Find the average temperature during the first $$8$$ hours of the day.c. Use the Trapezoidal Rule with $$4$$ equal subintervals to estimate$$\int_{0}^{8}T(H)dH$$.d. Find an expression for the rate at which the temperature is changing with respect to $$H$$.

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2. A population of algae grows in a pond. The rate of growth of the algae is proportional to the amount present at time $$t$$, so that $$\frac{dA}{dt}=kA$$ where $$k$$ is a positive constant and $$A(t)$$ is measured in tons.a. Find an expression for $$A(t)$$ in terms of $$t$$, given that the initial amount of algae is $$50$$ tons and that after $$2$$ years the amount has increased to $$80$$ tons.b. After how many years will the amount of algae reach $$200$$ tons?c. Suppose that algae-eating organisms are introduced into the pond and remove algae at a constant rate of $$30$$ tons per year. Write a differential equation that models the new situation, and determine how long it will take for the pond to become completely free of algae.

   • a. $$A(t)=50\left(\frac{8}{5}\right)^{t/2}$$ b. $$t=\frac{2\ln4}{\ln\frac{8}{5}}$$ c. $$\frac{dA}{dt}=kA-30$$ and $$t=\frac{1}{k}\ln\left(\frac{30}{30-50k}\right)$$ with $$k=\frac{1}{2}\ln\frac{8}{5}$$
   • a. $$A(t)=50\left(\frac{8}{5}\right)^{t/2}$$ b. $$t=\frac{2\ln4}{\ln\frac{8}{5}}$$ c. $$\frac{dA}{dt}=kA-30$$ and $$t=\frac{1}{k}\ln\left(\frac{30}{30-50k}\right)$$ with $$k=\frac{1}{2}\ln\frac{8}{5}$$