91Ó°ÊÓ

Bacteria Count The number \(N\) of bacteria in a refrigerated food is given by \(N(T)=10 T^{2}-20 T+600, \quad 2 \leq T \leq 20\) where \(T\) is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by \(T(t)=3 t+2, \quad 0 \leq t \leq 6\) where \(t\) is the time in hours. (a) Find the composition \((N \circ T)(t)\) and interpret its meaning in context. (b) Find the bacteria count after 0.5 hour. (c) Find the time when the bacteria count reaches 1500 .

The cost per unit in the production of an MP3 player is 60 dollars. The manufacturer charges 90 dollars per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by 0.15 dollars per MP3 player for each unit ordered in excess of 100 (for example, there would be a charge of 87 dollars per MP3 player for an order size of 120 ). (a) The table shows the profits \(P\) (in dollars) for various numbers of units ordered, \(x .\) Use the table to estimate the maximum profit. $$\begin{array}{|l|c|c|c|c|c|}\hline \text { Units, } x & 130 & 140 & 150 & 160 & 170 \\\\\hline \text { Profit, } P & 3315 & 3360 & 3375 & 3360 & 3315 \\\\\hline\end{array}$$ (b) Plot the points \((x, P)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(P\) as a function of \(x ?\) (c) Given that \(P\) is a function of \(x,\) write the function and determine its domain. (Note: \(P=R-C\) where \(R\) is revenue and \(C\) is cost.)

Find an equation of the line passing through the points. Sketch the line. $$(-8,0.6),(2,-2.4)$$

When you are given two functions \(f(x)\) and \(g(x),\) you can calculate \((f \circ g)(x)\) if and only if the range of \(g\) is a subset of the domain of \(f\).

A company produces a product for which the variable cost is 12.30 dollars per unit and the fixed costs are 98,000 dollars. The product sells for 17.98 dollars. 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: \(P=R-C\) ).

Beam Load The maximum load that can be safely supported by a horizontal beam varies jointly as the width of the beam and the square of its depth and inversely as the length of the beam. Determine the changes in the maximum safe load under the following conditions. A. The width and length of the beam are doubled. B. The width and depth of the beam are doubled.

The function \(F(y)=149.76 \sqrt{10} y^{5 / 2}\) estimates the force \(F\) (in tons) of water against the face of a dam, where \(y\) is the depth of the water (in feet). (a) Complete the table. What can you conclude from the table? $$\begin{array}{|l|l|l|l|l|l|}\hline y & 5 & 10 & 20 & 30 & 40 \\\\\hline F(y) & & & & & \\\\\hline\end{array}$$ (b) Use the table to approximate the depth at which the force against the dam is \(1,000,000\) tons. (c) Find the depth at which the force against the dam is \(1,000,000\) tons algebraically.

For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate \(=8-0.05(n-80), \quad n \geq 80\) where the rate is given in dollars and \(n\) is the number of people. (a) Write the revenue \(R\) for the bus company as a function of \(n\) (b) Use the function in part (a) to complete the table. What can you conclude? $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline n & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\\\\hline R(n) & & & & & & & \\\\\hline\end{array}$$

Write the standard form of the equation of the circle with the given characteristics. Endpoints of a diameter: \((0,0),(6,8)\)

Intercept Form of the Equation of a Line In Exercises \(81-86,\) use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts \((a, 0)\) and \((0, b)\) is \(\frac{x}{a}+\frac{y}{b}=1, a \neq 0, b \neq 0\) Point on line: \((1,2)\) \(x\) -intercept: \((c, 0)\) \(y\) -intercept: \((0, c), \quad c \neq 0\)