9: Math Models and Geometry
- Page ID
- 5011
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We are surrounded by all sorts of geometry. Architects use geometry to design buildings. Artists create vivid images out of colorful geometric shapes. Street signs, automobiles, and product packaging all take advantage of geometric properties. In this chapter, we will begin by considering a formal approach to solving problems and use it to solve a variety of common problems, including making decisions about money. Then we will explore geometry and relate it to everyday situations, using the problem-solving strategy we develop.
- 9.1: Use a Problem Solving Strategy (Part 1)
- In earlier chapters, you translated word phrases into algebraic expressions and word sentences into algebraic equations and solved some word problems that applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated. Now we'll develop a strategy you can use to solve any word problem.
- 9.2: Use a Problem Solving Strategy (Part 2)
- In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don't usually arise on an everyday basis, but they provide a good introduction to practicing the Problem Solving Strategy. Some number word problems ask you to find two or more numbers. Be sure to read the problem carefully to discover how all the numbers relate to each other.
- 9.3: Solve Money Applications
- Solving coin word problems is much like solving any other word problem. However, what makes them unique is that you have to find the total value of the coins instead of just the total number of coins. For coins of the same type, the total value can be found by multiplying the number of coins by the value of an individual coin. You may find it helpful to put all the numbers into a table to make sure they check.
- 9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 1)
- An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. If the sum of the measures of two angles is 180°, then they are supplementary angles. But if their sum is 90°, then they are complementary angles. We will adapt our Problem Solving Strategy for Geometry Applications. Since these applications will involve geometric shapes, it will help to draw a figure and label it with the information from the problem.
- 9.5: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 2)
- Triangles are named by their vertices. For any triangle, the sum of the measures of the angles is 180°. Some triangles have special names such as the right triangle which has one 90° angle. The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. To solve problems that use the Pythagorean Theorem, we will need to find square roots.
- 9.6: Use Properties of Rectangles, Triangles, and Trapezoids (Part 1)
- Many geometry applications will involve finding the perimeter or the area of a figure. The perimeter is a measure of the distance around a figure. The area is a measure of the surface covered by a figure. The volume is a measure of the amount of space occupied by a figure. A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L, and the adjacent side as the width, W.
- 9.7: Use Properties of Rectangles, Triangles, and Trapezoids (Part 2)
- Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of a triangle is one-half the base times the height. An isosceles triangle is a triangle with two sides of equal length is while a triangle that has three sides of equal length is an equilateral triangle. A trapezoid is four-sided figure with two sides that are parallel, the bases, and two sides that are not. The area of a trapezoid is one-half the height times the sum of the bases.
- 9.8: Solve Geometry Applications- Circles and Irregular Figures
- In this section, we will work on geometry applications for circles and irregular figures. To solve applications with circles, we use the properties of circles from Decimals and Fractions. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. To find the area of one of these irregular figures, we split it into figures whose formulas we know and then add the areas of the figures.
- 9.9: Solve Geometry Applications- Volume and Surface Area (Part 1)
- The surface area is a square measure of the total area of all the sides of a rectangular solid. The amount of space inside the rectangular solid is the volume, a cubic measure. The volume, V, of any rectangular solid is the product of the length, width, and height. To find the surface area of a rectangular solid, find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.
- 9.10: Solve Geometry Applications- Volume and Surface Area (Part 2)
- A sphere is the shape of a basketball, like a three-dimensional circle. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height h of a cylinder is the distance between the two bases. In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.
- 9.11: Solve a Formula for a Specific Variable
- For an object moving in at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula d = rt where d = distance, r = rate, and t = time. To solve a formula for a specific variable means to get that variable by itself with a coefficient of 1 on one side of the equation and all the other variables and constants on the other side. The result is another formula, made up only of variables.
Figure 9.1 - Note the many individual shapes in this building. (credit: Bert Kaufmann, Flickr)
Contributors and Attributions
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."