Chapter 4, Problem 8P

Auto Exhaust Emissions A study by the U.S. Office of Science and Technology in 1972 estimated the cost of reducing automobile emissions by certain percentages. Find an exponential model that captures the “diminishing returns” trend of these data shown in the table below.

Reduction in emissions (%)	Cost per car($)
50	45
55	55
60	62
65	70
70	80
75	90
80	100
85	200
90	375
95	600

To determine

To find: The exponential model that captures the “diminishing returns” trend of the data given.

Answer to Problem 8P

The exponential model that captures the “diminishing returns” trend of the given data is $y=2.41\times {1.058}^x$, where the variable $x$ represents the reduction in emission (%) and the variable $y$ represents the cost per car in ($).

Explanation of Solution

Given:

The data is as follows.

Reduction in emission (%)	Cost per car ($)
50	45
55	55
60	62
65	70
70	80
75	90
80	100
85	200
90	375
95	600

Let the variable $x$ be the reduction in emission (%) and $y$ be the cost per car in ( $ ).

Use online graphing calculator and draw the exponential function of the form $y=a{b}^x$ that gives the best fits of the data point as shown below in Figure 1.

From Figure 1, it is noticed that the exponential function $y=2.41\times {1.058}^x$ gives the best fit of the data point with the scatter plot.

Therefore, the exponential model that captures the “diminishing returns” trend of the given data is $y=2.41\times {1.058}^x$, where the variable $x$ represents the reduction in emission (%) and the variable $y$ represents the cost per car in ( $ ).