An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Assume the linear model fits the data well.
Calculate t for 2012: t = 2012 − 1995 = 17 .
Estimate retail sales in 2012: S ( 17 ) = 9.44 ( 17 ) + 84.182 = 244.662 ≈ 245 billion.
Predict the year when sales reach $241 billion: 241 = 9.44 t + 84.182 ⇒ t ≈ 16.612 ⇒ Y e a r = 1995 + 16 = 2011 .
The estimated retail sales in 2012 is 245 billions of dollars and the predicted year is 2011 .
Explanation
Assess the Linear Model First, we need to determine if the linear model is a good fit for the data. Since we don't have the scatter plot, we will assume, for the sake of this problem, that the function is a good model for the data. If the scatter plot showed a non-linear pattern, we would conclude that the linear model is not a good fit. However, without the plot, we proceed assuming it's a reasonable fit.
Calculate t Next, we estimate the retail sales in the U.S. in 2012. To do this, we need to find the value of t , which represents the number of years since 1995. We calculate t as follows: t = 2012 − 1995 = 17
Substitute t into the Model Now, we substitute t = 17 into the linear model S ( t ) = 9.44 t + 84.182 :
S ( 17 ) = 9.44 ( 17 ) + 84.182
Calculate S(17) Calculating the value: S ( 17 ) = 160.48 + 84.182 = 244.662
Round to Nearest Whole Number Rounding to the nearest whole number, we get: S ( 17 ) ≈ 245 So, the estimated retail sales in 2012 are approximately $245 billion.
Set S(t) = 241 Now, we need to predict the year in which retail sales will be 241 bi ll i o n . W ese t S(t) = 241 an d so l v e f or t : 241 = 9.44 t + 84.182 $
Isolate the Term with t Subtract 84.182 from both sides: 241 − 84.182 = 9.44 t 156.818 = 9.44 t
Solve for t Divide by 9.44: t = 9.44 156.818 ≈ 16.612
Calculate the Year Since t represents the number of years since 1995, we add t to 1995 to find the predicted year: Y e a r = 1995 + 16.612 ≈ 2011.612
Round Down to the Nearest Year Rounding down to the nearest whole number, we get the year 2011.
Final Answer Therefore, based on the linear model and the given data, the function is a good model for the data. The estimated retail sales in the U.S. in 2012 is approximately $245 billion, and the model predicts that retail sales will reach $241 billion in the year 2011.
Examples
Linear models can be used to predict sales trends in business. For example, a store owner might track monthly sales data and use a linear model to predict future sales. This can help them make decisions about inventory, staffing, and marketing. By understanding the trend, they can prepare for future demand and optimize their business operations. This approach is also used in broader economic forecasting to understand market trends and make informed investment decisions.
