# Section 2.3 – Modeling with First Order Equations

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. A 120-gallon tank initially contains 90 lb of salt dissolved in 90 gallons of water. Brine containing 2 lb/gal of salt flows into the tank at a rate of 4 gal/min, and the well-stirred mixture flows out of the tank at a rate of 3 gal/min. How much salt does the tank contain when it is full?

202 lbs of salt

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2. A cake is removed from an oven at 210$$^\circ$$ F and left to cool at room temperature, which is 70$$^\circ$$ F. After 30 min the temperature of the cake is 140$$^\circ$$ F. When will it be 100$$^\circ$$ F?

Approximately 66 minutes and 40 seconds.

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3. In a certain culture of bacteria, the number of bacteria increases sixfold in 10hrs. How long does it take for the population to double?

Approximately 3.87 hours

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4. A ball with mass 1kg is thrown upward with  initial velocity 20 m/s from the roof of a building 50 m high. A force due to the resistance of the air of $$v/10$$, where the velocity is measured in m/s, acts on the ball. Find the maximum height  above the ground that the ball reaches.

Approximately 68 m.
Video Errata: At the 40:33 mark, the answer should be approximately equal to 68 m. This is also noted in the video.

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5. In a typical Swedish summer day, temperature fluctuates sinusoidally between 10 (at midnight) and 30 (at noon) degrees Celsius. Assume at a party someone forgets a root beer with a temperature of 5 degrees at 10 pm, but finds it again at 2 am. What will be the temperature of the beer then, assuming a heat transfer coefficient of 0.1 per hour?

Approximately $$9.1^\circ$$ C.

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6. Consider a large lake formed by damming a river that initially holds 200 million liters of water. Because a nearby chemical plant uses the lake's water to clean its reservoirs, 1,000 liters of brine, each containing $$100(1 + \cos t)$$ kilograms of dissolved pollution, run into the lake every hour. Let's make the simplifying assumption that the mixture, kept uniform by stirring, runs out at the rate of 1,000 liters per hour and no additional spraying causes the lake to become even more contaminated. Find the amount of pollution $$p(t)$$ in the lake at any time $$t$$, and determine its variations in the long run.

$$p(t)=2\cdot10^7+100\cdot5\cdot10^{-6}\cos(t)+100\sin(t)-2\cdot10^7e^{-5\cdot10^{-6}t},$$ as t gets larger, $$p(t)\approx2\cdot10^7+100\sin(t).$$

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