Mark D.

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Calculus

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Question:

A single hose is used to fill a large swimming pool. It is determined that the volume of water in the pool V, can be represented as a function of time t as follows: V(t) = 100t + 30 Where t is in seconds and V is in liters. What is the flow rate of the hose in liters / second?

Mark D.

Answer:

Here we use the definition of the derivative. We know how many liters, V will be in the pool at any given second, t. We also know that the derivative represents the rate of change. The quantity (dV / dt), therefore, represents the change in water volume over time. In other words, it is the flow rate of water in liters / second. We compute dV / dt as follows: dV / dt = (dV/dt) (100t + 30) = 100 liters / second The flow rate of the hose is 100 liters / second.

Finance

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Question:

Today is April 11, 2017. You purchase 100 shares of Cracker Barrel (CBRL) common stock at a price of $52.40 per share. Your projections indicate that CBRL will rise to $57.40 one year from now, on April 11, 2018. You also anticipate receiving dividends in the amount of $3.75 per share during that time. If your predictions are true, what is your expected annual return, as a percentage? In dollars (to the nearest dollar)? Neglect tax effects.

Mark D.

Answer:

We first compute the expected annual return as a percentage. It's important to note that, for this part, the number of shares is irrelevant! The percentage return will be the same regardless of the number of shares purchased. % Return = Profit / Initial Investment We profit in two ways: from stock price appreciation, and from the collection of dividends. We receive dividends of $3.75, and our stock gains amount to ($57.40 - $52.40) = $5.00. Thus, we have total profit of ($3.75 + $5.00) = $8.75. % Return = $8.75 / $52.40 = .1670 = 16.7% From here, it is quite easy to compute the dollar return. We know that our profit is $8.75 per share purchased. Having purchased 100 shares, our total dollar profit will be ($8.75 * 100) = $875. We can also arrive at this answer by using the percentage return we computed. Our total initial investment is ($52.40 * 100) = $5240, and we expect a return of 16.7%. Therefore our total dollar profit will be ($5240 * .1670) = $875.08, with the $.08 difference being due to rounding. Thus, we again have a profit of $875 to the nearest dollar.

Physics

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Question:

A skydiver drops a penny from an F-16 jet at a height of 3000 meters. Neglecting effects of air resistance, at what speed does the penny hit the ground?

Mark D.

Answer:

We use the law of conservation of energy: Total Energy Final = Total Energy Initial. What types of energy do we have in the initial state, the instant the penny is dropped from 3000 meters? We have only potential energy due to gravity, PEg. Ei = PEg = mgh What types of energy do we have in the final state, the instant the penny hits the ground? We have only kinetic energy, KE. Ef = KE = (1/2)mv^2 Setting these two expressions equal to each other, we find that m cancels and we have: mgh = (1/2)mv^2 gh = (1/2)v^2 Solving for velocity: v = sqrt(2gh) = sqrt(2*9.8*3000) = 242.5 m/s This is equivalent to over 542 miles per hour!

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