91Ó°ÊÓ

In Exercises 39-46, determine the intervals over which the function is increasing, decreasing, or constant. \(f(x) = x^3 - 3x^2 + 2\)

TAXES State sales tax is based on retail price. An item that sells for \(\$189.99\) has a sales tax of \(\$11.40\). Find a mathematical model that gives the amount of sales tax \(y\) in terms of the retail price \(x\). Use the model to find the sales tax on a \(\$639.99\) purchase.

An overhead garage door has two springs, one on each side of the door (see figure). A force of 15 pounds is required to stretch each spring 1 foot. Because of a pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the combined lifting force applied to the door by the springs when the door is closed.

NEWTON'S LAW OF UNIVERSAL GRAVITATION: The gravitational attraction \(F\) between two objects of masses \(m_1\) and \(m_2\) is proportional to the product of the masses and inversely proportional to the square of the distance \(r\) between the objects.

LOGISTIC GROWTH: The rate of growth \(R\) of a population is jointly proportional to the size \(S\) of the population and the difference between \(S\) and the maximum population size \(L\) that the environment can support.

In Exercises 57-66, use a graphing utility to graph the function and approximate (to two decimal places) any relative minimum or relative maximum values. \(g(x) = x\sqrt{4-x}\)

In Exercises 67-74, find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) \(A\) varies directly as \(r^2\). \((A = 9 \pi when r = 3.)\)

In Exercises 67-70, find the value(s) of \(x\) for which \(f(x)=g(x)\). \(f(x)=\sqrt{x}-4\), \(g(x)=2-x\)

CONSUMER AWARENESS The suggested retail price of a new hybrid car is \(p\) dollars. The dealership advertises a factory rebate of \(\$2000\) and a \(10\%\) discount. (a) Write a function \(R\) in terms of \(p\) giving the cost of the hybrid car after receiving the rebate from the factory. (b) Write a function \(S\) in terms of \(p\) giving the cost of the hybrid car after receiving the dealership discount. (c) Form the composite functions \((R \circ S)(p)\) and \((S \circ R)(p)\) and interpret each. (d) Find \((R \circ S)(20,500)\) and \((S \circ R)(20,500)\). Which yields the lower cost for the hybrid car? Explain.

RESISTANCE In Exercises 77 and 78, use the fact that the resistance of a wire carrying an electrical current is directly proportional to its length and inversely proportional to its cross-sectional area. If #28 copper wire (which has a diameter of 0.0126 inch) has a resistance of 66.17 ohms per thousand feet, what length of #28 copper wire will produce a resistance of 33.5 ohms?