a) $10\sqrt10$ million b) 31.6 million; underestimate; 0.1 million (100,000)
Work Step by Step
a) The formula $E=5\sqrt x+34.1$ represents the number of elderly Americans ages 65-84 (in millions) x years after 2010. So, to find the number of elderly Americans in 2050 (40 years after 2010), plug 40 into x. The resulting equation becomes $E=5\sqrt 40+34.1$. However, before simplifying the radical, the problem asks for the increase in elderly Americans since 2010. In order to find the starting number of elderly Americans being compared in this scenario, plug 0 into x since there are 0 years since 2010 in the starting year 2010. The equation for 2010 becomes $E=5\sqrt 0+34.1=34.1$. In order to find the increase, or difference, in elderly Americans since 2010 to the year 2050, subtract the two newly created equations representing their respective years. $(5\sqrt 40+34.1)-34.1=5\sqrt 40$. Simplify the radical by figuring out what perfect square is a factor of 40. 4 has a perfect square of 2, and 4$\times$10 is 40. Multipy the 5 in front of the radical by the 2 and keep the remaining 10 inside the radical, since it cannot be simplified, to get $10\sqrt 10$ million elderly Americans. b) Using a calculator to simplify $10\sqrt10$ million results in the value 31.6 million when rounded to the nearest tenth. This value underestimates (lower than) the difference represented by the graph (65.8-34.1=31.7 million) by 0.1 million (100,000).