Algebra/Chapter 2/Exercises

A set of exercises related to concepts from Chapter 2.

Conceptual Questions

Q2.1 (Are they the Same?) Is “3 less than a number” the same thing as “the difference of 3 and a number”? What about "3 more than a number" and "the sum of 3 and a number"? Explain your reasoning in both cases.

Q2.2 (Coefficients) Does the expression x + 2 have any coefficients for x? What about the expression yx + 2, where any number can take the place of y? If so, identify the coefficient of x in each expression? If not, explain your reasoning.

Q2.3 (Anatomy of a Mathematical Expression) Look at the expression below. Define all of the simplified expression's terms, variables, coefficents, and constants.

$3x^{2}+4y+y+6$

Q2.4 (Constants and Variables) Letters will be given to represent various numbers. Decide if the following quantities should be referred to as variables or constants.

$a.\ T$, the temperature outside of your house.
$b.\ F$, the number of fingers on an average person's hand.
$c.\ P$, the price of a gallon of gas.
$d.\ L$, the number of leaves on a tree.
$e.\ S$, the number of sides on a rectangle.
$f.\ I$, the number of inches in a foot.
$g.\ Y$, the number of years since the last moon landing.
$h.\ D$, the number of donuts in an unopened box of a dozen.
$i.\ W$, the number of windows open on your computer screen.
$j.\ R$, the number of problems in this book you have attempted.
Q2.5 (Museum Admissions) The total cost to get admission to a museum is 25a + 10c + 8s for a adults, c children, and s seniors. How much does it cost for each adult, child, and senior respectively to get admission? If a family of two adults, three children, and one senior wants to gain admission to the museum, how much would they have to pay in total?

Q2.6 (Zero as a Constant Term?) An expression like x + y can be rewritten as x + y + 0. If this is the case, is it necessarily true that zero is a constant term for any given expression?

Q2.7 (Grammar in Mathematics) If mathematical expressions are analogous to "nouns" in a language, what part(s) of a mathematical expression is/are analogous to "verbs"? What part(s) are analogous to "conjunctions"?

Q2.8 (Identifying Mathematical Statements) Which of the following sentences are statements?

Q2.9 For the following problems, state whether the given statements are the same or different.

Exercises

Section 2.1

2.1 (Writing/Simplifying Expressions) Write an expression that best represents the following. Simplify whenever possible.

2.2 (Evaluating Expressions) Evaluate each expression for the given variable value.

$a.\ x+3,\ x=3$	$b.\ 10x-3,\ x=3$	$c.\ 4a-2,\ a=9$	$d.\ {\frac {-n-1}{3}},\ n=11$	$e.\ {\frac {-b-2}{7}},\ b=5$
$f.\$	$g.\$	$h.\$	$i.\$	$j.\$
$k.\$	$l.\$	$m.\$	$n.\$	$o.\$
$p.\$	$q.\$	$r.\$	$s.\$	$t.\$

Section 2.2

Problem 2.3 (Writing Mathematical Sentences) Write a mathematical sentence that best represents the following.

Reason and Apply

Problem 2.4 (Cutting Edge) A 12 ft long piece of rope was cut into two pieces of different lengths. Use one variable to represent the lengths of the two pieces.

Problem 2.5 (Play Ball!) The diameter of a basketball is approximately 4 times of that of a baseball. Express the diameter of a basketball in terms of the diamter of a baseball.

Problem 2.6 (Pocket Change) Suppose have d dimes and n nickels in your pocket. Write an expression which represents the total amount of money you have. Use this expression to figure out how much money you would have if you had 9 dimes and 7 nickels.

Problem 2.7 (Units of Temperature I) The formula

$C={\frac {5}{9}}(F-32)$

expresses the relationship between Farenheit temperature, F, and Celcius temperature, C. Use this equation to convert $50^{\circ }F$, $86^{\circ }F$, $32^{\circ }F$, $100^{\circ }F$, and $-50^{\circ }F$ to their equivalent temperature on the Celcius scale.

Problem 2.8 (Units of Temperature II) The formula

$K=C+273.15$

expresses the relationship between Celcius temperature, C, and Kelvin temperature, K. Use this equation to convert $-273.15^{\circ }C$, $30^{\circ }C$, and $100^{\circ }F$ to their equivalent temperature on the Kelvin scale.

Problem 2.9 (Chemical Formula for Sugar) The chemical formula for glucose (sugar) is $C_{6}H_{12}O_{6}$. This formula means there are 12 hydrogen atoms for every 6 carnon atoms and 6 oxygen atoms in each molecule of glucose. If x represents the number of atoms in oxygen in a pound of sugar, express the number of hydrogen atoms in the the same pound of sugar.

Problem 2.10 (Building Blocks) Look at the arrangements of building blocks below. How many blocks will appear in diagram 17?

Problem 2.11 (Triangles in Polygons) In a triangle, there are three sides. We can obviously observe from this that this contains 1 triangle. In a quadrilateral, there are four sides. We can observe from this that two non-overlapping triangles can be made out of this by dividing it along its corners. In a pentagon, there are five sides. We can observe from this that three non-overlapping trianges can be made out of this by dividing it along its corners. Using this information, how many non-overlapping triangles can you make out of a decagon (10-sided polygon) by dividing it along its corners?

Problem 2.12 (Product of Consecutive Numbers) Two numbers are consecutive if they follow each other in numerical order. For example, the numbers 4 and 5 are consecutive because 5 comes after 4. What would be an algebraic representation of the product of two such numbers?

Problem 2.13 (Odd Numbers) Write an expression that represents the nth odd number, O. (First odd number is 1, Second odd number is 3, and so fort1) Afterwards, use this expression to find the 143rd odd number.

Problem 2.14 (Magic Trick) Choose any number. Add 3 onto the number, then multiply the result by 2. Subtract the chosen number, then subtract 4, and then subtract the chosen number again. The number you end with is 2, isn't it? Why does this trick work?

Problem 2.15 (Exponentially Exciting) For each of the following, determine the first whole number x, greater than 1, for which the second expression is larger than the first.

$a.\ x^{3},\ 3^{x}$
$b.\ x^{4},\ 4^{x}$
$c.\ x^{5},\ 5^{x}$
$c.\ x^{6},\ 6^{x}$

Problem 2.16 ($x^{n}$ vs. $n^{x}$) On the basis of your answers to Problem 2.15, make a conjecture that appears to be true about the two expressions $x^{n}$ and $n^{x}$, where n = 3, 4, 5, 6, 7, .... and x is a whole number greater than 1.

Problem 2.17 (Weight-Loss Points) Several weight-loss programs assign points to prepared or packaged foods that take into account of the food's fat F, carbohydrate C, protein P, and fiber B content in grams. The point value for a given food item can be represented by the following expression:

$W={\frac {F}{4}}+{\frac {C}{9}}+{\frac {P}{10}}-{\frac {B}{12}}$

Determine the point value of one serving of the item having the nutrition facts on the left.

Problem 2.18 (Adjusted Poverty Threshold) The adjusted poverty threshold for a single person between 1999 and 2013 can be approximated by the formula

$y=2.719x^{2}+196.1x+8718$

where x=0 corresponds to 1999, x=1 corresponds to 2000, and so forth, and where y is the average adjusted poverty threshold. According to the model, what was the average adjusted poverty threshold in 2005? In 2012?

Problem 2.19 (Period of a Pendulum) The period t, in seconds, of the swing of a pendulum is given by

$t=2\pi {\sqrt {\frac {L}{32}}}$

where L is the length of the pendulum in feet. Find the period of a pendulum 8 feet long.

Problem 2.20 (Change in Length) Consider the triangle below, with sides of length s. Find the perimeter of the triangle if we increase the lengths of the sides by 5. Find the perimeter if we double the lengths of the sides.

Problem 2.21 (Metal Wire) A metal wire of length x is bent into a square. Express the length of a side of the square in terms of x.

Problem 2.22 (Area of a Rectangle) A rectangle has an area of 24 $in.^{2}$ and a length b in inches. What does the expression ${\frac {24}{b}}$ represent and what are its units of measurement? What quantity does the expression $2(b+{\frac {24}{b}})$ represent?

Problem 2.23 (Area of a Right Triangle) Derive an expression which can represent the area of a right triangle with base b and height h. (Hint: Bisect a rectangle along its diagonal.)

Problem 2.24 (Vegetable Garden) A rectangular vegetable garden is 12 meters long and 20 meters wide. Surrounding the garden is a gravel path of width w.
a. Write an expression that can be used to find the outer perimeter of the gravel path.
b. If you measure w to be 3 meters, what is the outer perimeter of the path?

Problem 2.25 (Racetrack) An Olympic racetrack is made up of two straight sides, each measuring 84.39 meters in length, and two semi-circular curves with a radius of 36.5 meters as pictured. The track has a width of w.
a. Write an expression that can be used to find the outer perimeter of the racetrack. (Remember that the perimeter of a circle is $2\pi r$)
b. If you measure that the width of the track is 1.22, what is the outer perimeter of the path?

Problem 2.26 (Volumes of Prisms) A prism consists of two paralell polygonall face ends of equal shape. A shape's volume is how much space it occupies. Take the rectangular prism below, for example.

Derive an expression which can represent the volume of the following prisms, and then calculate its volume.

Problem 2.27 (Difference of Squares)

$x$	$y$	$(x+y)(x-y)$	$x^{2}-y^{2}$

a. Choose two distinct values for x and y, and then fill in the first row for the table above.

b. Compare the results you got for the two expressions. What do you think the results from part a tell you about the difference of two squares?

c. Fill in the remaining rows of the table for different values of x and y, including negative numbers. Do you think your conjecture from part (b) is correct? Explain.

Problem 2.28 (Inequalities) Determine what sign values on $x$ and $y$ would make the following statements true.

$a.\ xy>0$	$b.\ x^{2}y>0$	$c.\ {\frac {x}{y}}<0$
$d.\ -{\frac {x}{y}}<0$	$e.\ -{\frac {x^{2}}{y}}>0$	$f.\ {\frac {y^{3}}{x^{2}}}>0$