Question

the percentage of the adult population that smokes can be modeled by the function ( p(x) ) whose graph is shown below. the input ( x ) is years after 1955.

in what year did the percentage of smokers start dropping below 20%?

the year that the percentage of smokers dropped below 20% is ( square ).

Explanation:

Step1: Analyze the graph

The input \(x\) is years after 1955. We need to find the \(x\) - value when \(P(x)<20\).
Looking at the graph, when \(P(x) = 20\), we can assume a linear relationship. Let's assume two - point form. If we assume the line passes through \((0,35)\) (assuming \(P(0)=35\)) and \((40,0)\) (since when \(x = 40\), \(P(x)=0\)). The equation of the line using the two - point formula \(y - y_1=\frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\) where \((x_1,y_1)=(0,35)\) and \((x_2,y_2)=(40,0)\) is \(P(x)=35-\frac{35}{40}x=35-\frac{7}{8}x\).

Step2: Solve for \(x\) when \(P(x) = 20\)

Set \(P(x)=20\), then \(20 = 35-\frac{7}{8}x\).
Subtract 35 from both sides: \(20 - 35=-\frac{7}{8}x\), so \(- 15=-\frac{7}{8}x\).
Multiply both sides by \(-\frac{8}{7}\): \(x=\frac{120}{7}\approx17.14\).

Since \(x\) represents years after 1955, the year is \(1955 + 17=1972\) (rounding up because when \(x = 17\), \(P(x)>20\) and when \(x = 18\), \(P(x)<20\))

Answer:

1972