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Math Topics

Applied Calculus

Applications of Integration

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1
Area of Regions in the Plane

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2
Consumer's and Producer's Surplus

The demand and supply for a particular type of chemical glassware is modeled by \[D(x)=-0.001x^2+154\] and \[S(x)=0.0002x^2+0.03x+100\] respectively, where \(x\) is the number of glassware units (in thousands) demanded and supplied, and both \(D(x)\) and \(S(x)\) are the glassware unit prices in dollars.

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3
Annuities and Money Streams

Revenue flows continuously into a retail operation at the constant rate of \(f(t)=128\text{,}000\) dollars per year. As the money is received, it is placed into a non-interest-earning account. Determine how much money has accumulated in the operationâ€™s account at the end of a \(3\)-year time period.

Suppose that at the end of each month, for \(6\) months, \(\$100\) is put into an account paying \(6\%\) annual interest compounded continuously. Except for the last \(\$100\), which earns no interest at all, each \(\$100\) earns interest over a different period of time. Using the continuous compounding formula \(A=Pe^{rt}\), find the total amount of money in the account at the end of \(6\) months.

Rental income from a piece of property, upon which there is an indefinite lease, flows at the rate of \(\$70\text{,}000\) per year. The income is invested immediately into an account paying \(6.5\%\) annual interest compounded continuously. Find and interpret the present value of the flow.

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4
Differential Equations

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5
Applications of Differential Equations

The value of real estate in a city is increasing, without limitations, at a rate proportional to its current value. A piece of property in the city appraised at \(\$240\text{,}000\) four months ago appraises at \(\$242\text{,}000\) today. At what value will the property be appraised \(12\) months from now?

The temperature of a machine when it is first shut down after operating is \(220^\circ\text{C}\). The surrounding air temperature is \(30^\circ\text{C}\). After \(20\) minutes, the temperature of the machine is \(160^\circ\text{C}\). Find a function that gives the temperature of the machine at any time, \(t\), and then find the temperature of the machine \(30\) after it is shut down.

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6
Probability

A stateâ€™s weather service provides a continuously running \(120\)-second recorded telephone message about weather conditions throughout the state. If a person calls and gets connected to the message he or she hears the message at any point in the \(120\)-second run. If a person calls and gets connected, what is the probability that he or she will hear at most \(20\) seconds of the message before it repeats?

A machine is adjusted to dispense \(6\) ounces of coffee into a \(7\)-ounce cup. The amount of coffee dispensed is normally distributed mean \(\mu=6\) ounces and standard deviation \(\sigma=0.5\) ounces. Find the probability that a randomly selected cup is filled with between \(5.5\) and \(5.8\) ounces of coffee.