In the year 1985, a house was valued at $113,000. By the year 2005, the value had appreciated exponentially to $155,000. What was the annual growth rate between 1985 and 2005 (Rourid your answer to two decimal places.) Assume that the value continued to grow by the same percentage. What was the value of the house in the year 2010? (Round your answer to the nearest dolar.)

Answer:The growth rate is 1.02; the value of the house in 2010 is $185,388Step-by-step explanation:This is an exponential growth equation, therefore, it follow the standard form:[tex]y=a(b)^x[/tex]where y is the value of the house after a certain number of years,a is the initial value of the house,b is the growth rate, andx is the year number.We are going to make this easy on ourselves and call year 1985 year 0.  Therefore, is year 1985 is year 0, then year 2005 is year 20, and year 2010 is year 25.  We will make these the x coordinates in our coordinate pairs.(0, 113000) and (20, 155000)Filling into our standard form using the first coordinate pair will give us the initial value of the house at the start of our problem:[tex]113000=a(b)^0[/tex]Anything raised to the 0 power is equal to 1, so113000 = a(1) anda = 113000Now we will use that value of a along with the second pair of coordinates and solve for b, the growth rate you're looking for:[tex]155000=113000(b)^{20}[/tex]Start by dividing both sides by 113000 to get a decimal:[tex]1.371681416=b^{20}[/tex]To solve for b, we have to undo that power of 20 by taking the 20th root of b.  Because this is an equation, we have to take the 20th root of both sides:[tex]\sqrt[20]{1.371681416}=\sqrt[20]{b^{20}}[/tex]The 20th root and the power of 20 undo each other so all we have left on the right is a b, and taking the 20th root on your calculator of the decimal on the left gives you:b = 1.0159 which rounds tob = 1.02  This is our growth rate.Now we can use this growth rate and the value of a we found to write the model for our situation:[tex]y=113000(1.02)^x[/tex]If we want to find the value of the house in the year 2010 (year 25 to us), we sub in a 25 for x and do the math:[tex]y=113000(1.02)^{25}[/tex]Raise 1.02 to the 25th power and get:y = 113000(1.640605994) and multiply to get a final value ofy = $185,388