Chapter 6

For Activities 1 through \(12,\) write an equation or differential equation for the given information. The cost \(c\) to fill a gas tank is directly proportional to the number of gallons \(g\) the tank will hold.

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. $$ \frac{d y}{d x}=2 x $$

Write a sentence of interpretation for the probability statement in the context of the given situation. \(P(x \geq 5)=0.46,\) where the random variable \(x\) is the length, in minutes, of a telephone call made on a computer software technical support line.

For Activities 1 through \(4,\) use numerical estimation to evaluate the improper integral. Show the numerical estimation table. \(\int_{0}^{\infty} 3 e^{-0.2 t} d t ;\) set \(t=5,\) increment \(\times 5,\) estimate to one decimal place.

Write an equation or differential equation for the given information. The marginal cost of producing window panes (that is, the rate of change of \(\operatorname{cost} c\) with respect to the number of units produced) is inversely proportional to the number of panes \(p\) produced.

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. $$ \frac{d y}{d x}=0.5 x^{2}+2 x $$

Write a sentence of interpretation for the probability statement in the context of the given situation. \(P(s>30)=0.58,\) where \(s\) is tomorrow's closing price, in dollars, of a share of Microsoft stock.

Determine whether the statement is true or false. Explain. If \(x\) is a random variable with a uniform density function for \(0 \leq x \leq 1\), its cumulative distribution function is $$ F(x)=\left\\{\begin{array}{ll} 0 & \text { for } x<0 \\ x & \text { for } 0 \leq x \leq 1 \\ 1 & \text { for } x>1 \end{array}\right. $$

a. Write the flow rate of the income stream. b. Calculate the 5 -year future value c. Calculate the 5 -year present value. Company B showed a profit of \(\$ 1.8\) million last year. The CEO of the company expects the profit to increase by 0.02 million dollars each year over the next 5 years and the profits will be continuously invested in an account bearing a \(4.75 \%\) APR compounded continuously.

Write an equation or differential equation for the given information. Barometric pressure \(p\) is changing with respect to altitude \(a\) at a rate that is proportional to the altitude.

Determine whether the statement is true or false. Explain. The value of \(k\) that makes $$ G(t)=\left\\{\begin{array}{ll} k e^{-t} & \text { when } t \geq 0 \\ 0 & \text { when } t<0 \end{array}\right. $$ an exponential density function is \(k=e\).

Write a sentence of interpretation for the probability statement in the context of the given situation. \(P(2 \leq t<4)=0.15,\) where \(t\) is the number of inches of rain that New Orleans receives, on average, during the month of March.

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. $$ \frac{d y}{d x}=6 x^{2} y $$

a. Write the flow rate of the income stream. b. Calculate the 5 -year future value c. Calculate the 5 -year present value. Company C showed a profic of \(\$ 1.8\) million last year. The CEO of the company expects the profit to decrease by \(7 \%\) each year over the next five years and the profits will be continuously invested in an account bearing a \(4.75 \%\) APR compounded continuously.

For Activities 1 through \(4,\) use numerical estimation to evaluate the improper integral. Show the numerical estimation table. \(\int_{-\infty}^{3} 2 e^{x} d x\) set \(x=-10,\) increment \(\times 4,\) estimate to three decimal places.

Write an equation or differential equation for the given information. The rate of change of the cost \(c\) of mailing a first-class letter with respect to the weight of the letter is constant.

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. $$ \frac{d y}{d x}=7 x(5-y) $$

Determine whether the statement is true or false. Explain. I'he cumulative distribution function of a uniform distribution function is a piecewise-defined linear function.

Write a sentence of interpretation for the probability statement in the context of the given situation. \(P(d<72)=0.34,\) where the random variable \(d\) is the distance, in feet, between any two cars on a certain two-lane highway.

a. Write the flow rate of the income stream. b. Calculate the 5 -year future value c. Calculate the 5 -year present value. Company D showed a profit of \(\$ 1.8\) million last year. The CEO of the company expects the profit to decrease by 0.04 million dollars each year over the next 5 years and the profits will be continuously invested in an account bearing a \(4.75 \%\) APR compounded continuously.

For Activities 1 through \(4,\) use numerical estimation to evaluate the improper integral. Show the numerical estimation table. \(\int_{-\infty}^{3}-2 e^{x} d x ;\) set \(x=-4,\) increment \(\times 2,\) estimate to the nearest integer.

Write an equation or differential equation for the given information. Ice thickens with respect to time \(t\) at a rate that is inversely proportional to its thickness \(T\).

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. $$ \frac{d y}{d x}=-x $$

Write a sentence of interpretation for the probability statement in the context of the given situation. \(P(a \geq 2)=0.25,\) where \(a\) is the age, in years, of a car rented from Hertz at the Los Angeles airport on \(12 / 28 / 2013 .\)

A supply function is given. a. Write the units of measure for the input and output variables of the supply function. b. Write a sentence of interpretation for each point given. The function \(S\) gives the number of pizzas (in hundreds) supplied by producers at a market price of \(p\) dollars per pizza. (5,16)\(;(16,24)\)

a. Write the units of measure for the input and output variables of the demand function. b. Write a sentence of interpretation for each point given. The function \(D\) gives the quantity of rice (in million pounds) purchased by consumers at a market price of \(p\) cents per pound. (16,5)\(;(160,0.05)\)

a. Write the flow rate of the income stream. b. Calculate the 5 -year future value c. Calculate the 5 -year present value. Company E showed a profit of \(\$ 1.8\) million last year. The CEO of the company expects the profit to decrease by 0.04 million dollars each year over the next 5 years and the profits will be continuously invested in an account bearing a \(4.75 \%\) APR compounded continuously.

For Activities 5 through \(16,\) evaluate the improper integral. $$ \int_{0.36}^{\infty} 9.6 x^{-0.432} d x= $$

Write an equation or differential equation for the given information. The Verhulst population model assumes that a population \(P\) in a country will be increasing with respect to time \(t\) at a rate that is jointly proportional to the existing population and to the remaining amount of the carrying capacity \(C\) of that country.

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. $$ \frac{d y}{d x}=\frac{y}{x} $$

Write a sentence of interpretation for the probability statement in the context of the given situation. \(P(0 \leq x<1.5)=0.9,\) where \(x\) is the waiting time, in hours, for a patient to see a doctor at a medical clinic.

A supply function is given. a. Write the units of measure for the input and output variables of the supply function. b. Write a sentence of interpretation for each point given. The function \(S\) gives the quantity of paint (in thousand gallons) supplied by producers when paint sells for \(p\) dollars per gallon. (12,36)\(;(19,52)\)

a. Write the units of measure for the input and output variables of the demand function. b. Write a sentence of interpretation for each point given. The function \(D\) gives the number of single-person aircraft ordered when a single-person aircraft sells for \(p\) thousand dollars. (64,560,000)\(;(320,2000)\)

a. Write the flow rate of the income stream. b. Calculate the 5 -year future value c. Calculate the 5 -year present value. Company F showed a profit of \(\$ 1.8\) million last year. The CEO of the company expects the profit to remain the same each year over the next 5 years and the profits will be continuously invested in an account bearing a \(4.75 \%\) APR compounded continuously.

For Activities 5 through \(16,\) evaluate the improper integral. $$ \int_{0.3}^{\infty} 5 x^{-0.4} d x $$

Write an equation or differential equation for the given information. The rate of change with respect to time \(t\) of the amount \(A\) that an investment is worth is proportional to the amount in the investment.

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. $$ \frac{d y}{d x}=10 x y^{-1} $$

A supply function is given. a. Write the units of measure for the input and output variables of the supply function. b. Write a sentence of interpretation for each point given. When coffee beans sell for \(p\) dollars per pound, producers supply \(q\) million pounds. (5,9)\(;(15,40)\)

a. Write the units of measure for the input and output variables of the demand function. b. Write a sentence of interpretation for each point given. When coffee beans sell for \(p\) dollars per pound, consumers demand \(q\) million pounds. (5,16)\(;(15,3)\)

For Activities 5 through \(16,\) evaluate the improper integral. $$ \int_{5}^{\infty} 5\left(0.36^{x}\right) d x $$

Write an equation or differential equation for the given information. The rate of change in the height \(h\) of a tree with respect to its age \(a\) is inversely proportional to the tree's height.

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. $$ \frac{d y}{d x}=\frac{-1}{x} $$

A supply function is given. a. Write the units of measure for the input and output variables of the supply function. b. Write a sentence of interpretation for each point given. Publishers will supply \(q\) thousand cconomics texts at a market price of \(p\) dollars per text. (57,14)\(;(295,1000)\)

Sara Lee For the year ending June \(30,2009,\) the revenue of the Sara Lee Corporation was \(\$ 12.88\) billion. Assume that Sara Lee's revenue will increase by \(5 \%\) per year and that beginning on July \(1,2009,3.5 \%\) of the revenue was invested each year (continuously) at an APR of \(5 \%\) compounded continuously. (Source: Hoover's Online Guide) a. Write the flow rate equation. b. What is the future value of the investment at the end of the year \(2013 ?\)

a. Write the units of measure for the input and output variables of the demand function. b. Write a sentence of interpretation for each point given. When Boy Scout popcorn sells for \(p\) dollars per tin, consumers demand \(q\) thousand tins. (5,200)\(;(7.50,180)\)

For Activities 5 through \(16,\) evaluate the improper integral. $$ \int_{5}^{\infty}\left[5\left(0.36^{x}\right)+5\right] d x $$

Write an equation or differential equation for the given information. In a community of \(N\) farmers, the number \(x\) of farmers who own a certain tractor changes with respect to time \(t\) at a rate that is jointly proportional to the number of farmers who own the tractor and to the number of farmers who do not own the tractor.

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. $$ \frac{d y}{d x}=e^{0.05 x} e^{-0.05 y} $$

Match each given situation to a possible graph of its density function. Explain. A random number generator is used to choose a real number between 0 and \(100 .\) The random variable \(x\) is the number chosen.

Waiting Time A traffic light on campus remains red for 30 seconds at a time. A car arrives at that light and finds it red. Assume that the waiting time \(t\) seconds at the light follows a uniform density function \(f\). a. Calculate the car's chances of waiting at least 10 seconds at the red light. b. Calculate the probability of waiting no more than 20 seconds at the red light. c. What is the average expected wait time?