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Average Price The average prices \(p\) (in thousands of dollars) of a new mobile home in the United States from 1990 to 2002 (see figure) can be approximated by the model $$ p(t)= \begin{cases}0.182 t^{2}+0.57 t+27.3, & 0 \leq t \leq 7 \\ 2.50 t+21.3, & 8 \leq t \leq 12\end{cases} $$ where \(t\) represents the year, with \(t=0\) corresponding to 1990. Use this model to find the average price of a mobile home in each year from 1990 to 2002 . (Source: U.S. Census Bureau)

Road Grade You are driving on a road that has a \(6 \%\) uphill grade (see figure). This means that the slope of the road is \(\frac{6}{100}\). Approximate the amount of vertical change in your position if you drive 200 feet.

Cost, Revenue, and Profit A company produces a product for which the variable cost is \(\$ 12.30\) per unit and the fixed costs are \(\$ 98,000\). The product sells for \(\$ 17.98\). Let \(x\) be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost \(C\) as a function of the number of units produced. (b) Write the revenue \(R\) as a function of the number of units sold. (c) Write the profit \(P\) as a function of the number of units sold. (Note: \(P=R-C\) )

You are given the dollar value of a product in 2005 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value \(V\) of the product in terms of the year t. (Let \(t=5\) represent 2005.) 2005 Value $$ \$ 2540 $$ Rate \$125 decrease per year

Height of a Balloon A balloon carrying a transmitter ascends vertically from a point 3000 feet from the receiving station. (a) Draw a diagram that gives a visual representation of the problem. Let \(h\) represent the height of the balloon and let \(d\) represent the distance between the balloon and the receiving station. (b) Write the height of the balloon as a function of \(d\). What is the domain of the function?

Think About It When plotting points on the rectangular coordinate system, is it true that the scales on the \(x\) - and \(y\)-axes must be the same? Explain.

Think About It Suppose you correctly enter an expression for the variable \(y\) on a graphing utility. However, no graph appears on the display when you graph the equation. Give a possible explanation and the steps you could take to remedy the problem. Illustrate your explanation with an example.

Cost, Revenue, and Profit A rooling contractor purchases a shingle delivery truck with a shingle elevator for \(\$ 36,500\). The vehicle requires an average expenditure of \(\$ 5.25\) per hour for fuel and maintenance, and the operator is paid \(\$ 11.50\) per hour. (a) Write a linear equation giving the total \(\operatorname{cost} C\) of operating this equipment for \(t\) hours. (Include the purchase cost of the equipment.) (b) Assuming that customers are charged \(\$ 27\) per hour of machine use, write an equation for the revenue \(R\) derived from \(t\) hours of use. (c) Use the formula for profit \((P=R-C)\) to write an equation for the profit derived from \(t\) hours of use. (d) Use the result of part (c) to find the break-even point-that is, the number of hours this equipment must be used to yield a profit of 0 dollars.

Use the Quadratic Formula to solve the equation. $$ (z+6)^{2}=-2 z $$

Rental Demand A real estate office handles an apartment complex with 50 units. When the rent per unit is \(\$ 580\) per month, all 50 units are occupied. However, when the rent is \(\$ 625\) per month, the average number of occupied units drops to 47 . Assume that the relationship between the monthly rent \(p\) and the demand \(x\) is linear. (a) Write the equation of the line giving the demand \(x\) in terms of the rent \(p\). (b) Use this equation to predict the number of units occupied when the rent is \(\$ 655\). (c) Predict the number of units occupied when the rent is \(\$ 595\).