The line graph shows the cost of inflation in some country. What cost​ $100 in 1999 would cost the amount shown by the graph in subsequent years. Below are two mathematical models for the data shown in the graph. In each​ formula, C represents the cost x years after 2010 of what cost​ $100 in 1999. Answer parts​ (a) through​ (c).

Question

Model 1   Cequals1.9xplus120.4
Model 2   Cequals0.03x squaredplus1.8xplus120.6
The Cost of Inflation
2010
2012
2014
2016
2018
2020
120
125
130
135
140
145
150
Year
Cost (dollars)

A line graph titled The Cost of Inflation has a horizontal axis labeled Year from 2010 to 2020 in increments of 2 years and a vertical axis labeled Cost (dollars) from 120 to 150 in increments of 5. The graph consists of six plotted points connected by line segments from left to right. The coordinates of the plotted points are the following, with the year listed first and the cost in dollars listed second: 2010, 120; 2012, 125; 2014, 129; 2016, 130; 2018, 135; 2020, 140.
Question content area bottom
Part 1
a. Use the graph to estimate the cost in 2018 of what cost​ $100 in 1999.
​$
  
135
Part 2
b. Use model 1 to determine the cost in 2018. How well does this describe your estimate from part​ (a)?
The cost in 2018 was approximately ​$
  
136.
​(Round to the nearest dollar as​ needed.)
Part 3
Select the correct choice below and fill in the answer box to complete your choice.
​(Round to the nearest dollar as​ needed.)
A.
The formula appears to overestimate the cost by ​$
  
enter your response here​, when compared to the graph.​ Thus, the formula does not describe well the estimate from part​ (a).
B.
The formula appears to underestimate the cost by ​$
  
enter your response here​, when compared to the graph.​ Thus, the formula describes well the estimate from part​ (a).
C.
The formula appears to underestimate the cost by ​$
  
enter your response here​, when compared to the graph.​ Thus, the formula does not describe well the estimate from part​ (a).
D.
The formula appears to overestimate the cost by ​$
  
enter your response here​, when compared to the graph.​ Thus, the formula describes well the estimate from part​ (a).

c. Use model 2 to determine the cost in 2018. How well does this describe your estimate from part​ (a)?
The cost in 2018 was approximately ​$
  
135.
​(Round to the nearest dollar as​ needed.)

Answer 1

Let's break down each part of your question.

Part 1: Cost in 2018 from the Graph

From the graph, you have already estimated the cost in 2018 of what cost $100 in 1999 as $135.

Part 2: Using Model 1

The formula for Model 1 is: \[ C = 1.9x + 120.4 \] where \( C \) is the cost in dollars and \( x \) is the number of years since 2010.

To find the cost in 2018, we first calculate \( x \):

• For 2018: \( x = 2018 - 2010 = 8 \)

Now plug \( x = 8 \) into Model 1: \[ C = 1.9(8) + 120.4 \] \[ C = 15.2 + 120.4 \] \[ C = 135.6 \]

Rounded to the nearest dollar, this becomes: \[ C \approx 136 \]

Comparison with Estimate from Part 1

The estimate from the graph was $135, and Model 1 gives approximately $136.

Part 3: Determine Over/Underestimation

Now we calculate the difference:

• Difference = Model 1 result - Graph estimate = 136 - 135 = 1

Since Model 1 gives a result that is higher than the graph estimate, we refer to it as an overestimation.

Conclusion for Part 3

The correct choice is: A. The formula appears to overestimate the cost by $1, when compared to the graph. Thus, the formula does not describe well the estimate from part (a).

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Part 4: Using Model 2

The formula for Model 2 is: \[ C = 0.03x^2 + 1.8x + 120.6 \]

Again, using \( x = 8 \): \[ C = 0.03(8^2) + 1.8(8) + 120.6 \] Calculating \( 8^2 \):

• \( 8^2 = 64 \)

Now substituting: \[ C = 0.03(64) + 14.4 + 120.6 \] \[ C = 1.92 + 14.4 + 120.6 \] \[ C = 136.92 \] Rounded to the nearest dollar, this becomes: \[ C \approx 137 \]

Comparison with Estimate from Part 1

Now we need to find out how this compares with the estimate of $135.

Difference

• Difference = Model 2 result - Graph estimate = 137 - 135 = 2.

Since Model 2 also provides a higher estimate than the graph, it too represents an overestimation.

Conclusion for Part 4

You can conclude that the cost in 2018 according to Model 2 is approximately $137.

Also you can write: The formula appears to overestimate the cost by $2, when compared to the graph. Thus, the formula does not describe well the estimate from part (a).