Chapter 4, Problem 3PS

Distinct Vision The endpoints of the interval over which distinct vision is possible are called the near point and far point of the eye (see figure). With increasing age, these points normally change. The table shows the approximate near points y (in inches) for various ages x (in years).

(a) Use the regression feature of a graphing utility to find a quadratic model for the data. Use the graphing utility to plot the data and graph the model in the same viewing window.

(b) Find a rational model for the data. Take the reciprocals of the near points to generate the points $(x,1/y)$. Use the regression feature of the graphing utility to find a linear model for the data. The resulting line has the form

$$\frac{1}{y}=ax+b$$

Solve for y. Use the graphing utility to plot the original data and graph the model in the same viewing window.

(c) Use the table feature of the graphing utility to construct a table showing the predicted near point based on each model for each of the ages in the original table. How well do the models fit the original data?

(d) Use both models to estimate the near point for a person who is 25 years old. Which model

is a better fit?

(e) Do you think either model can be used to predict the near point for a person who is 70

years old? Explain.