136 Chapter 2 Linear Equations and Inequalities in One Variable �� ������������������������ Formulas and Applications from Geometry The Columbus Circle in New York City, built in 1905, was the first traffic circle in the United States. It was renovated a century later in 2005. The distance around the inner circle measures approximately 673 ft. The distance around the outer circle is approximately 1364 ft. Knowing these values, we can find the radius of each circle and also the area of each circle. The formulas that enable us to solve this problem are provided in this section. (Source: http://en.wikipedia.org/wiki/Columbus_Circle) Formulas A mathematical formula is an equation that expresses the relationship between two or more quantities. These quantities are represented by variables. Mathematical formulas enable us to find valuable information if we know some specific values of some of the variables in the formula. In Section 1.3, we evaluated algebraic expressions; some of these were mathematical formulas. Some of the commonly used formulas are for calculating the area and perimeter of geometric shapes, converting degrees Fahrenheit to degrees Celsius, and calculating the amount of money in a savings account. Some of these formulas are listed within this section. Evaluating formulas is exactly like evaluating algebraic expressions. ���������������������� Evaluating Formulas Step 1: Substitute known values in place of the appropriate variables. Step 2: Simplify the resulting expression or solve the resulting equation. �� ������������������������ ������������������ Use the formula and the given values to solve for the missing variable. 1a. The formula A = P(1 + r)t is used to calculate the amount of money in an account at the end of t years if P dollars are initially invested at an annual interest rate, r. Find the total value, A, if $10,000 is invested for 5 yr at 4.5% annual interest. Solution 1a. A = P(1 + r)t State the formula. A = 10,000(1 + 0.045)5 Let P = 10,000, r = 4.5% = 0.045, and t = 5. A = 10,000(1.045)5 Add the expression in parentheses. A = 10,000(1.246182) Evaluate the exponential expression. A = $12,461.82 Multiply. In 5 yr, the total value in the account will be $12,461.82. 1b. The formula F = 9 5 C + 32 is used to convert degrees Celsius (C ) to degrees Fahrenheit (F ). Find the value of C if F = 63°. Solution 1b. F = 9 5 C + 32 State the formula. 63 = 9 5 C + 32 Replace F with 63. 5(63) = 5a9 5 C + 32b Multiply each side by the LCD, 5. 315 = 5a9 5 Cb + 5(32) Simplify and apply the distributive property. 315 = 9C + 160 Simplify. Objective 1 ▶ Evaluate formulas. id d i thi ti ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Evaluate formulas. 2. Use perimeter, area, and circumference formulas. 3. Solve formulas for a specified variable. 4. Solve problems involving complementary and supplementary angles. 5. Solve problems involving straight and vertical angles. 6. Solve problems involving triangles. 7. Troubleshoot common errors.
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