Chapter 5

The area of a rectangular rug is 48 square feet. The length of the rug is 2 feet longer than the width. What are the dimensions of the rug?

Imaginary numbers are often used in electrical engineering. The impedance \(Z\) (measured in ohms) of a circuit measures the resistance of a circuit to alternating current \((\mathrm{AC})\) electricity. For two \(\mathrm{AC}\) circuits connected in series, the total impedance is the sum \(Z_{1}+Z_{2}\) of the individual impedances. Find the total impedance if \(Z_{1}=5 i\) ohms and \(Z_{2}=8 i\) ohms.

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{8+2 i}{1+3 i} $$

The difference in the lengths of the sides of two squares is 1 meter. The difference in the areas of the squares is 13 square meters. What are the lengths of the sides of the squares?

In certain circuits, the total impedance \(Z_{T}\) is given by the formula \(Z_{T}=\frac{Z_{1} Z_{2}}{Z_{1}+Z_{2}} .\) Find \(Z_{T}\) when \(Z_{1}=-3 i\) and \(Z_{2}=4 i\)

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{3+i}{3-i} $$

The perimeter of a rectangle is 24 feet. The area of the rectangle is 32 square feet. Find the dimensions of the rectangle.

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{5-2 i}{5+2 i} $$

The endpoints of a diameter of a circle are \((0,0)\) and \((8,4) .\) a. Write an equation of the circle and draw its graph. b. On the same set of axes, draw the graph of \(x+y=4\) c. Find the common solutions of the circle and the line. d. Check the solutions in both equations.

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{12+3 i}{3 i} $$

a. On the same set of axes, sketch the graphs of \(y=x^{2}-4 x+5\) and \(y=2 x+2\) b. Does the system of equations \(y=x^{2}-4 x+5\) and \(y=2 x+2\) have a common solution in the set of real numbers? Justify your answer. c. Find the solution set.

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{8+6 i}{2 i} $$

a. On the same set of axes, sketch the graphs of \(y=x^{2}+5\) and \(y=2 x\) b. Does the system of equations \(y=x^{2}+5\) and \(y=2 x\) have a common solution in the set of real numbers? Justify your answer. c. Does the system of equations \(y=x^{2}+5\) and \(y=2 x\) have a common solution in the set of complex numbers? If so, find the solution.

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{\frac{1}{5}-\frac{1}{5} i}{\frac{5}{3}-4 i} $$

The graphs of the given equations have three points of intersection. Use an algebraic method to find the three solutions of this system of equations: $$ \begin{array}{l}{y=x^{3}-2 x+1} \\ {y=2 x+1}\end{array} $$

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{1+i}{\frac{1}{2}-\frac{1}{2} i} $$

A soccer ball is kicked upward from ground level with an initial velocity of 52 feet per second. The equation \(\mathrm{h}(t)=-16 t^{2}+52 t\) gives the ball's height in feet after \(t\) seconds. To the nearest tenth of a second, during what period of time was the height of the ball at least 20 feet?

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{3 i}{\pi+\frac{1}{2} i} $$

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{\frac{1}{7}+i}{\frac{5}{i}} $$

The profit function, in thousands of dollars, for a company that makes graphing calculators is \(\mathrm{P}(x)=-5 x^{2}+5,400 x-106,000\) where \(x\) is the number of calculators sold in the millions. a. Graph the profit function \(\mathrm{P}(x)\) b. How many calculators must the company sell in order to make a profit?

Impedance is the resistance to the flow of current in an electric circuit measured in ohms. The impedance, \(Z,\) in a circuit is found by using the formula \(Z=\frac{V}{I}\) where \(V\) is the voltage (measured in volts) and \(I\) is the current (measured in amperes). Find the impedance when \(V=1.8-0.4 i\) volts and \(I=-0.3 i\) amperes.

Find the current that will flow when \(V=1.6-0.3 i\) volts and \(Z=1.5+8 i\) ohms using the formula \(Z=\frac{V}{I} .\)