6. Since 1992, the value of homes in a neighborhood has doubled every 16 years. The value of one home in the neighborhood was \( \$ 136,500 \) in 1992 a. What is the value of this home, in dollars, in the year 2000? Explain your reasoning. b. Write an equation that represents the growth in housing value as a function of time in \( t \) years since 1992. c. Write an equation that represents the growth in housing value as a function of time in \( d \) decades since 1992. d. Use one of your equations to find the value of the home, in dollars, 1.5 decades after 1992. (From Unit 4, Lesson 4.) Algebra 2
Solution
Step 1 ：a. Since 2000 is 8 years after 1992, divide 16 years by 2 to get the half-growth period: \(\frac{16}{2} = 8\) years. Multiply the home value in 1992 by 2 for half growth: \(136,500 \times 2 = \$ 273,000\).
Step 2 ：b. \(V(t) = 136,500 \times 2^{\frac{t}{16}}\) where \(V(t)\) is the value of the home in dollars and \(t\) is the number of years since 1992.
Step 3 ：c. Since there are 10 years in a decade, replace \(t\) with \(10d\): \(V(d) = 136,500 \times 2^{\frac{10d}{16}}\), where \(V(d)\) is the value of the home in dollars and \(d\) is the number of decades since 1992.
Step 4 ：d. Substitute 1.5 decades into the equation c: \(V(1.5) = 136,500 \times 2^{\frac{10 \times 1.5}{16}}\), which gives \(V(1.5) = 136,500 \times 2^{\frac{15}{16}} \approx \$ 265,967.66\).
