91Ó°ÊÓ

A mathematical model can be used to describe the relationship between the number of feet a car travels once the brakes are applied, \(y,\) and the number of seconds the car is in motion after the brakes are applied, \(x .\) A research firm collects the following data: $$\begin{array}{cc}\begin{array}{c}x, \text { seconds in motion } \\\\\text { after brakes are applied } \end{array} & \begin{array}{c}y, \text { feet car travels } \\\\\text { once the brakes are applied }\end{array} \\\\\hline 1 & 46 \\\\\hline 2 & 84 \\\\\hline 3 & 114\end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=6 .\) Describe what this means.

Use a graphing utility to sketch the region determined by the constraints. Then determine the maximum value of the objective function subject to the contraints. Objective Function \(\quad z=6 x+8 y\) Constraints \(\quad x \geq 0, y \geq 0\) \(x+2 y \leq 6\)

In Exercises \(19-30,\) solve each system by the addition method. \(3 x=4 y+1\) \(3 y=1-4 x\)

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&x+2 y \leq 4\\\&y \geq x-3\end{aligned} $$

Solve each system by the method of your choice. $$\begin{aligned} &3 x^{2}+4 y^{2}=16\\\ &2 x^{2}-3 y^{2}=5 \end{aligned}$$

Write the partial fraction decomposition of each rational expression. $$\frac{5 x^{2}-6 x+7}{(x-1)\left(x^{2}+1\right)}$$

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&x+y \leq 4\\\&y \geq 2 x-4\end{aligned} $$

Solve each system by the method of your choice. $$\begin{aligned} &x+y^{2}=4\\\ &x^{2}+y^{2}=16 \end{aligned}$$

Use a graphing utility to sketch the region determined by the constraints. Then determine the maximum value of the objective function subject to the contraints. Objective Function \(\quad z=30 x+20 y\) $$\begin{array}{ll} \text { Constraints } & x \geq 0, y \geq 0 \\ & 2 x+y \leq 14 \\ & 3 x+y \leq 18 \end{array}$$

In Exercises \(19-30,\) solve each system by the addition method. \(5 x=6 y+40\) \(2 y=8-3 x\)