6.2 Formulas and Modeling 307 In Examples 1 and 2, we used an equation to represent real-life phenomena. By doing so, we say we have created a mathematical model or simply a model to represent the situation. A model may be a formula, a single equation, or several equations considered simultaneously. In some exercises, when we are asking you to write a model, we will mark the exercise with the word MODELING. Did You Know? L mc E mc 2 2 = = Many of you recognize the formula E mc2 = used by Albert Einstein in 1912 to describe his groundbreaking theory of relativity. In the formula, E is energy, m is mass, and c is the speed of light. In his theory, Einstein hypothesized that time was not absolute and that mass and energy were related. Einstein’s 72-page handwritten manuscript shows that Einstein toyed with using an L rather than an E to represent energy. He wrote L mc ,2 = and then scratched out the L and replaced it with an E to produce the equation E mc .2 = Yisrail Galili at Jerusalem’s Hebrew University said the L might have been a lambda, a Greek letter sometimes used in functions related to energy. He said it might also be related to Joseph-Louis Lagrange, an eighteenth-century mathematician. m Albert Einstein Example 2 Volume of a Cylinder The volume of a cylinder is determined by the formula, V r h, 2 π = where r is the radius and h is the height of the cylinder. An ice cream container has the shape of a cylinder and has a radius of 2.5 inches and a height of 5.5 inches. Determine the volume of the ice cream container. Use the π key on a calculator when evaluating the formula and round the answer to the nearest hundredth. Solution As discussed in Section 5.4, π is an irrational number that is approximately equal to 3.14. When we use π in a calculation, we will use the π key on our calculator to obtain a more accurate answer. To determine the volume of the ice cream container, we substitute the appropriate values into the volume of a cylinder formula and evaluate. V r h, (2.5) (5.5) (6.25)(5.5) (34.375) 107.99 2 2 π π π π = = = = ≈ Thus, the volume of the ice cream container is about 107.99 cubic inches. 7 Now try Exercise 59 Example 1 Simple Interest Mateo invests $3000 in a certificate of deposit (CD) that pays 4.3% (or 0.043) simple interest for 2 years. The simple interest formula is interest principal = × i prt rate time, or . × = a) Use the simple interest formula to determine the amount of simple interest that Mateo earns on the CD. b) Determine the total value of the CD, including the interest earned, at the end of the 2 years. Solution a) The principal, p, is the amount of money invested, or $3000. The rate, r, is the simple interest rate written as a decimal, or 0.043. The time, t, is 2 years. We substitute these values into the simple interest formula and evaluate. i prt 3000 (0.043)(2) 258 = = = Thus, Mateo earns $258 simple interest. b) The total value of the CD is the original principal, $3000, plus the simple interest earned, $258. Thus, the total value of the CD after 2 years is $3000 $258, or $3258. + 7 Now try Exercise 53 Our next example involves the use of a geometry formula which we will study in Section 8.4. Library of Congress Prints and Photographs Division [LC-USZ62-60242]
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