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Using a Formula

A right circular cylinder with radius rr and height hh has the surface area SS (in square units) given by the formula S=2πr(r+h)S=2\pi r(r+h). See Figure 3. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of π\pi.

Right Circular Cylinder

Solution

Evaluate the expression 2πr(r+h)2\pi r(r+h) for r=6r=6 and h=9h=9.

S=2πr(r+h)=2π(6)[(6)+(9)]=2π(6)(15)=180π\begin{align*} S &= 2\pi r(r+h) \\ &= 2\pi (6)[(6)+(9)] \\ &= 2\pi (6)(15) \\ &= 180\pi \\ \end{align*}

The surface area is 180π180\pi square inches.

Case

The above case, including the title and solution, is attributed to College Algebra 2e. Page 19.

Q&A​

What are the set of rational numbers?​

Page 8, College Algebra 2e

The set of rational numbers is written as {mn | m and n are integers and n≠0}\{\frac{m}{n} \text{ | } m \text{ and } n \text{ are integers and } n \not = 0 \} . Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

  • a terminating decimal:158=1.875\frac{15}{8} = 1.875, or
  • a repeating decimal: 411=0.36363636…=0.36‾\frac{4}{11} = 0.36363636\mathellipsis=0.\overline{36}

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

What are irrational numbers?​

Page 9, College Algebra 2e

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even 32\frac{3}{2} but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown. {h | h is not a rational number}\{h \text{ | } h \text{ is not a rational number}\}

What are real numbers?​

Page 10, College Algebra 2e

Given any number n, we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one- to-one correspondence. We refer to this as the real number line as shown in Figure 1.

The Real Number Line

Evaluating Algebraic Expressions​

Page 17, College Algebra 2e

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as x+5,43πr3x+5,\frac{4}{3} \pi r^3, or 2m3n2\sqrt{2m^3n^2} . In the expression x+5x+5, 55 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

3(−3)5=(−3)⋅(−3)⋅(−3)⋅(−3)⋅(−3)x5=x⋅x⋅x⋅x⋅x(2⋅7)3=(2⋅7)⋅(2⋅7)⋅(2⋅7)(yz)3=(yz)⋅(yz)⋅(yz)\begin{align*} {3} (-3)^5 &= (-3)\cdot(-3)\cdot(-3)\cdot(-3)\cdot(-3) &\quad x^5 &= x \cdot x \cdot x \cdot x \cdot x \\ (2\cdot7)^3 &= (2\cdot7) \cdot (2\cdot7) \cdot (2\cdot7) &\quad (yz)^3 &= (yz)\cdot(yz)\cdot(yz) \\ \end{align*}

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

What is an equation?​

Page 19, College Algebra 2e

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation 2x+1=72x+1=7 has the solution 33 because when we substitute 33 for xx in the equation, we obtain the true statement 2(3)+1=72(3)+1=7.

What is a formula?​

Page 19, College Algebra 2e

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area AA of a circle in terms of the radius rr of the circle: A=πr2A=\pi r^2. For any value of rr, the area AA can be found by evaluating the expression πr2\pi r^2.

Glossary​

Equation​

https://en.wikipedia.org/wiki/Equation

An equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign ==.

Last accessed

January 01, 2024

Expression​

https://en.wikipedia.org/wiki/Expression_(mathematics)

An expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.

Last accessed

January 01, 2024

Irrational Number​

https://en.wikipedia.org/wiki/Irrational_number

All the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers.

Among irrational numbers are the ratio π\pi of a circle's circumference to its diameter, Euler's number ee, the golden ratio ϕ\phi, and the square root of two.[1] In fact, all square roots of natural numbers, other than of perfect squares, are irrational.

Last accessed

January 01, 2024

Rational Number​

https://commons.wikimedia.org/wiki/Category:Rational_numbers

A rational number is any number that can be expressed as the quotient or fraction pq\frac{p}{q} of two integers, with the denominator qq not equal to zero. Since qq may be equal to 1, every integer is a rational number. This category represents all rational numbers, that is, those real numbers which can be represented in the form: a b ab\frac{a}{b} ...where aa and bb are integers and bb is not equal to zero. All integers are rational, including zero.

Last accessed

December 26, 2023

Real Number​

https://en.wikipedia.org/wiki/Real_number

The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the real numbers are called irrational numbers.

Real numbers can be thought of as all points on a line called the number line or real line, where the points corresponding to integers (..., −2, −1, 0, 1, 2, ...) are equally spaced.

Last accessed

January 01, 2024

Surface Area​

https://en.wikipedia.org/wiki/Surface_area

The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies.

Last accessed

January 01, 2024

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