To find the linear equation [tex]\( p(n) = m n + b \)[/tex] that describes the price [tex]\( p \)[/tex] based on the number of shirts [tex]\( n \)[/tex] sold, we need to follow these steps:
1. Determine the slope [tex]\( m \)[/tex]:
The slope of a linear equation can be found using the formula:
[tex]\[
m = \frac{p_2 - p_1}{n_2 - n_1}
\][/tex]
Substituting the given points [tex]\((n_1, p_1) = (9000, 117)\)[/tex] and [tex]\((n_2, p_2) = (43000, 15)\)[/tex], we get:
[tex]\[
m = \frac{15 - 117}{43000 - 9000} = \frac{-102}{34000}
\][/tex]
Simplifying this, we find:
[tex]\[
m \approx -0.003
\][/tex]
(The slope is rounded to three decimal places)
2. Determine the y-intercept [tex]\( b \)[/tex]:
The y-intercept can be found using the slope-intercept form of the linear equation [tex]\( p(n) = m n + b \)[/tex]. We already know one of the points (9000, 117) and the slope [tex]\( m \)[/tex].
Substitute [tex]\( n_1 = 9000 \)[/tex], [tex]\( p_1 = 117 \)[/tex], and [tex]\( m = -0.003 \)[/tex] into the equation [tex]\( p(n) = m n + b \)[/tex]:
[tex]\[
117 = -0.003 \cdot 9000 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
117 = -27 + b
\][/tex]
[tex]\[
b = 144
\][/tex]
Finally, we have both the slope [tex]\( m = -0.003 \)[/tex] and the y-intercept [tex]\( b = 144 \)[/tex]. Hence, the linear equation can be expressed as:
[tex]\[
p(n) = -0.003 n + 144
\][/tex]
This is the linear model describing the relationship between the number of shirts sold and the price per shirt.
