91Ó°ÊÓ

Break-Even Analysis In Exercises 57 and 58 , find the sales necessary to break even \((R=C)\) for the total cost \(C\) of producing \(x\) units and the revenue \(R\) obtained by selling \(x\) units. (Round to the nearest whole unit.) $$C=5.5 \sqrt{x}+10,000, \quad R=3.29 x$$

Consumer Surplus and Producer Surplus In Exercises \(61-64,\) (a) graph the systems representing the consumer surplus and producer surplus for the supply and demand equations and (b) find the consumer surplus and producer surplus. $$\begin{array}{ll}{\text { Demand }} & {\text { Supply }} \\ {p=50-0.5 x} & {p=0.125 x}\end{array}$$

Consumer Surplus and Producer Surplus In Exercises \(61-64,\) (a) graph the systems representing the consumer surplus and producer surplus for the supply and demand equations and (b) find the consumer surplus and producer surplus. $$\begin{array}{ll}{\text { Demand }} & {\text { Supply }} \\ {p=100-0.5 x} & {p=25+0.1 x}\end{array}$$

Consumer Surplus and Producer Surplus In Exercises \(61-64,\) (a) graph the systems representing the consumer surplus and producer surplus for the supply and demand equations and (b) find the consumer surplus and producer surplus. $$\begin{array}{ll}{\text { Demand }} & {\text { Supply }} \\ {p=140-0.00002 x} & {p=80+0.00001 x}\end{array}$$

Log Volume Two rules for estimating the number of board feet in a log include the Doyle Log Rule and the Scribner Log Rule. (A board foot is a unit of measure for lumber equal to a board 1 foot square and 1 inch thick.) For a 16 -foot log, the Doyle Log Rule is modeled by \(V_{1}=(D-4)^{2}, 5 \leq D \leq 40,\) and the Scribner Log Rule is modeled by \(V_{2}=0.79 D^{2}-2 D-4,5 \leq D \leq 40\) where \(D\) is the diameter (in inches) of the log and \(V\) is its volume (in board feet).

Geometry The perimeter of a triangle is 180 feet. The longest side of the triangle is 9 feet shorter than twice the shortest side. The sum of the lengths of the two shorter sides is 30 feet more than the length of the longest side. Find the lengths of the sides of the triangle.

Geometry In Exercises 65 and \(66,\) find the dimensions of the rectangle meeting the specified conditions. The perimeter is 56 meters and the length is 4 meters greater than the width.

Writing the Partial Fraction Decomposition , write the partial fraction decomposition of the rational expression. Then assign a value to the constant \(a\) to to check the result algebraically and graphically. $$\frac{1}{x(x+a)}$$

Ticket Sales For a concert event, there are \(\$ 30\) reserved seat tickets and \(\$ 20\) general admission tickets. There are 2000 reserved seats available, and fire regulations limit the number of paid ticket holders to \(3000 .\) The promoter must take in at least \(\$ 75,000\) in ticket sales. Find and graph a system of inequalities describing the different numbers of tickets that can be sold.

Writing Describe two ways of solving for the constants in a partial fraction decomposition.