172 Chapter 2 Linear Equations in Two Variables and Functions Writing a Linear Model Example 1 Buffalo, New York, had 2 ft (24 in.) of snow on the ground before a snowstorm. During the storm, snow fell at an average rate of in./hr. a. Write a linear equation to compute the total snow depth y after x hr of the storm. 5 8 b. Graph the equation. c. Use the equation to compute the depth of snow after 8 hr. d. If the snow depth was 31.5 in. at the end of the storm, determine how long the storm lasted. Solution: a. The constant or base amount of snow before the storm began is 24 in.The 58 5 8 rate of change is given by in. of snow per hour. If m is replaced by and b is replaced by 24, we have the linear equation y mx b y 5 8 x 24 b. The equation is in slopeintercept form, and the corresponding graph is shown in the figure. c. 5 8 x 24 5 8 182 24 y Snow Depth (in.) y x 8 Substitute . Solve for y. y 5 24 in. y 29 Snow Depth Versus Time 40 35 y 58 x 24 30 25 5 in. 20 15 10 5 0 0 2 4 6 8 10 12 14 16 18 20 Time (hr) 8 hr The snow depth was 29 in. after 8 hr.The corresponding ordered pair is (8, 29) and can be confirmed from the graph. d. Substitute . 5 8 x 24 5 8 x 24 y 31.5 y 31.5 Multiply by 8 to clear fractions. Clear parentheses. Solve for x. 8131.52 8 a5 8 x 24b 252 5x 192 60 5x 12 x The storm lasted for 12 hr.The corresponding ordered pair is (12, 31.5) and can be confirmed from the graph. Avoiding Mistakes A mathematical model may be reasonable only for specific values of the input variable. In Example 1, the model y x 24 is reasonable for 0 x 12. For x values less than 0, the snow storm hadn’t started. For x values greater than 12, the storm had ended.We say that “model breakdown” 58 occurs for values of x outside the reasonable range.
miller_intermediate_algebra_4e_ch1_3
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