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Homework for Chapter 1

Exercises on lines

Exercise 1.1 : For each pair of points A( ) and B( ) find (i) and in going from to , (ii) the slope of the line joining and , (iii) the equation of the line joining A and B in the form , and (iv) the distance from to :

(a) A(2,0), B(4,3);

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(b) A(1,-1), B(0,2);

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(d) A(-2,3), B(4,3);

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(e) A(-3,-2), B(2,3);

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(f) A(0.01,-0.01), B(-0.01,0.05);

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Exercise 1.2 : Graph each of the following lines, after changing to the form ,
and also find the y-intercept and x-intercept:

code to generate graph paper

(a) ;

[Maple Plot]

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(b) ;

[Maple Plot]

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[Maple Plot]

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(d) ;

[Maple Plot]

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(e)

[Maple Plot]

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Exercises on Circles

Exercise 1.3 : Find the equation of the circle of radius 3 centered at:

(a) (0,0);

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(b) (5,6);

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(d) (0,3);

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(e) (0,-3);

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(f) (3,0).

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Exercise 1.4 : Graph the circles:

(a) ;

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(b) ;

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Story problems

Exercise 1.5: Let x stand for temperature in degrees Celsius (centigrade) , and let y stand for temperature in degrees Fahrenheit . A temperature of (freezing of water) corresponds to , and a temperature of (boiling of water) corresponds to . Find the equation of the line that relates temperature Fahrenheit y to temperature Celsius x. Graph the line, and find the y- and x-intercepts. What is the practical meaning of the intercepts?

Exercise 1.6: A car rental firm has the following charges for a certain type of car: $25 per day with 100 free miles included, $0.15 per mile for more than 100 miles. Suppose you want to rent a car for one day, and you know you'll use it for more than 100 miles. What is the equation relating the cost y to the number of miles x that you drive the car?
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Exercise 1.7: An instructor gives a 100-point final exam, and decides that a score 90 or above will be a grade of 4.0, a score of 40 or below will be a grade of 0.0, and between 40 and 90 the grading will be linear. Let x be the exam score, and let y be the corresponding grade. Find a formula of the form which applies to scores x between 40 and 90.
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Exercise 1.8: A photocopy store advertises the following prices: 5 cents per copy for the first 20 copies, 4 cents per copy for the through copy, and 3 cents per copy after the copy. Let x be the number of copies, and let y be the total cost of photocopying. (a) Graph the cost as x goes from 0 to 200 copies. (b) Find the equation in the form that tells you the cost of making x copies when x is more than 100.

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Exercise 1.9: In the Kingdom of Zog the tax system works as follows. Someone who earns less than 100 gold coins per month pays no tax. Someone who earns between 100 and 1000 golds coins pays tax equal to 10 % of the amount over 100 gold coins that he or she earns. Someone who earns over 1000 gold coins must hand over to the King all of the money earned over 1000 in addition to the tax on the first 1000. (a) Draw a graph of the tax paid y versus the money earned x, and give formulas for y in terms of x in each of the regions , , and . (b) Suppose that the King of Zog decides to use the second of these line segments (for ) for as well. Explain in practical terms what the King is doing, and what the meaning is of the y-intercept.

Exercise 1.10: Market research tells you that if you set the price of an item at $1.50, you will be able to sell 5000 items; and for every 10 cents you lower the price below $1.50 you will be able to sell another 1000 items. Let x be the number of items you can sell, and let P be the price of an item.

(a) Express P linearly in terms of x, in other words, express P in the form .

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(b) Express x linearly in terms of P.

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More line problems

Exercise 1.11: Do the following points all lie on a line? If so, find the equation for this line.

, , , , , . Explain.

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Exercise 1.12: If the point ( ) lies on the line with slope passing through the point (2,5), find .

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Exercise 1.13: Does the point (3, -2) lie on the line through the points (8,0) and (-7,-6)? Explain.

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Exercise 1.14: Use the slopes to determine whether the points (7,-1), (10,1), and (6,7) are the vertices of a right triangle.
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Exercise 1.16: Find a point such that the line through and meets the line segment connecting and at the point of the distance from to and at the midpoint of the segment . Hint: Find the ``1/3 point'' first.

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