Chapter 5

An efficiency expert finds that the average number of defective items produced in a factory is approximately linearly related to the average number of hours per week the employees work. If the employees average 34 hours of work per week, then an average of 678 defective items are produced. If the employees average 45 hours of work per week, then an average of 834 defective items are produced. (Round to the nearest tenth) (a) Write an equation relating the average number of defective items \(d\) and the average number of hours \(h\) the employees work. (b) What would be the average number of defective items if the employees average 40 hours per week? (c) According to this relationship, how many hours per week would the employees need to average in order to reduce the average number of defective items to \(500 ?\) Do you think this is a practical goal? Explain.

A doctor uses a treadmill to administer cardiac stress tests to his patients. The treadmill is 5.5 feet long with a front end that can be raised to a maximum height of 10 inches. Find the maximum grade of the treadmill.

Define \(x\) - and \(y\) -intercepts in two ways: (a) In terms of the graph of an equation (b) In terms of an algebraic solution to the equation

A slanted roof on a house rises 2.8 meters over a horizontal distance of 6.2 meters. Find the grade (or pitch) of the roof.

Consider the equation \(c=3 n+2\) (a) Sketch the graph of this equation using the horizontal axis for the \(n\) values and the vertical axis for the \(c\) values. (b) Use the graph to describe what happens to the values of \(c\) as \(n\) changes. (c) Sketch the graph of this equation using the horizontal axis for the \(c\) values and the vertical axis for the \(n\) values. (d) Use the graph to describe what happens to the values \(n\) as \(c\) changes. (e) Does the way we choose to label the axes affect the appearance of the graph? Does the way we choose to label the axes affect the relationship between the variables?

An exceptionally steep portion of a ski slope, called the Perils of Pauline, falls 632 meters over a horizontal distance of 925 meters. Find the grade of the Perils of Pauline.

Simplify the given expression. $$x^{2}\left(x^{3}\right)\left(x^{2}\right)^{3}$$

This exercise discusses the relationship between the slopes of perpendicular lines. (a) Sketch the graphs of \(y=2 x+4\) and \(y=-\frac{1}{2} x+4\) on the same coordinate system. (b) On the basis of your graph, does it appear that these lines are perpendicular? (c) What is the relationship between the slopes of these two lines? (d) It is a fact that the slopes of perpendicular lines are negative reciprocals of each other (provided that neither of the lines is vertical). What is the slope of a line perpendicular to the line whose equation is \(y=\frac{2}{5} x+7 ?\)

An engincer has specificd that a sewage pipe for a certain building must have a grade of \(3 \% .\) How much vertical clearance must the construction crew leave for a sewage pipe that must carry waste a horizontal distance of 200 feet?

Simplify the given expression. $$\left(2 x^{2}\right)(5 x y)\left(-y^{2}\right)+x^{3} y^{3}$$

Solve each of the following problems algebraically. Be sure to label what the variable represents. The width of a rectangle is 4 less than \(\frac{2}{3}\) its length. If the perimeter of the rectangle is 3 times the length, find its dimensions.

The conveyor belt on a certain assembly line has a grade of \(3.2 \% .\) If the belt carries items through a vertical distance of \(12.8 \mathrm{ft}\), how long is the belt?

Simplify the given expression. $$\left(2 x^{2}\right)\left(5 x y-y^{2}\right)-x^{2} y^{2}$$

Solve each of the following problems algebraically. Be sure to label what the variable represents. A beaker contains \(40 \mathrm{ml}\) of a \(60 \%\) alcohol solution. What percentage of alcohol solution must be added to produce \(70 \mathrm{ml}\) of a \(45 \%\) solution?

How could you use the idea of slope to show that the three points \((-1,-2)\) \((2,0),\) and \((5,2)\) all lie on a straight line?

Solve the following problem algebraically. Be sure to label what the variable represents. Lamont has invested \(\$ 1,300\) in a savings account that pays \(4 \%\) annual interest. At what interest rate must an additional \(\$ 800\) be invested to produce \(\$ 100\) per year in interest?

Solve for \(x: \frac{x}{3}-\frac{x-1}{4}=\frac{x+1}{2}\)

Combine: \(\frac{x}{3}-\frac{x-1}{4}+\frac{x+1}{2}\)

Solve the following problem algebraically. Be sure to label what the variable represents. Tamika leaves point \(A\) at 10: 00 A.M. traveling due east at 60 kph. One-half hour later, Ramon leaves the same location traveling due west at \(70 \mathrm{kph}\). At what time will they be \(257.5 \mathrm{km}\) apart?