Question 1192979: Topics In Contemporary Math
Linear Growth
4) The average amount a television viewer spent on merchandise, apparel, and snacks when
watching a Super Bowl game is shown.
Year 2005 2007 2009 2011 2012
Amount $38.35 $56.04 $57.27 $59.33 $63.87
a) If we consider 2005 as the base year, write an explicit formula for 𝑃𝑛, the amount of
money spent on merchandise, apparel, and snacks when watching a Super Bowl game for
n years after 2005.
b) How much would the model predict is spent in 2022?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **a) Find the Explicit Formula**
1. **Calculate the Average Annual Increase:**
* Find the differences in spending between consecutive years:
* 2007 - 2005: $56.04 - $38.35 = $17.69
* 2009 - 2007: $57.27 - $56.04 = $1.23
* 2011 - 2009: $59.33 - $57.27 = $2.06
* 2012 - 2011: $63.87 - $59.33 = $4.54
* Calculate the average annual increase:
* (17.69 + 1.23 + 2.06 + 4.54) / 4 = $6.38
2. **Write the Explicit Formula:**
* Since we're assuming linear growth, the formula for the amount spent (Pn) in year n is:
* Pn = P0 + r * n
* where:
* Pn is the amount spent in year n
* P0 is the initial amount spent (in 2005) = $38.35
* r is the average annual increase = $6.38
* n is the number of years after 2005
* **Therefore, the explicit formula is:**
* Pn = $38.35 + $6.38 * n
**b) Predict Spending in 2022**
* Calculate the number of years after 2005:
* 2022 - 2005 = 17 years
* Use the formula to predict spending in 2022:
* P17 = $38.35 + $6.38 * 17
* P17 = $38.35 + $108.46
* P17 = $146.81
**The model predicts that $146.81 would be spent on merchandise, apparel, and snacks when watching a Super Bowl game in 2022.**
**Important Note:** This model assumes a constant linear increase in spending each year. In reality, spending patterns may not follow a perfectly linear trend.
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