A company wants to know what price they should charge to make the maximum profit on their inexpensive digital watches. They have discovered that if they sell the watches for $1, they can sell 7 million of them, but if they sell them for $2, they can only sell 6 million. Let x be the number of million watches sold at p dollars per watch. Assume that the relationship between x and p is linear and find an equation that gives that relationship. Remember that revenue is price times number of items sold: R = xp . Solve the equation from the previous part for x and substitute the result into the equation R = xp in place of x ( R is in million of dollars). Graph the equation from previous part on a coordinate system where the horizontal axis is the p -axis and the vertical axis is the R -axis. The highest point on the graph will give you the maximum revenue. What is the maximum revenue?
A)Maximum revenue = $12,000,000 at $6 per watch.
B)Maximum revenue = $33,600,000 at $14 per watch.
C)Maximum revenue = $16,000,000 at $4 per watch.

Question

Quizplus · Accepted Answer

Given the linear relationship between the price $p$ and the number of watches sold $x$ (in millions), we can use the two points provided ($p=1$, $x=7$ and $p=2$, $x=6$) to find the slope of the line: $\frac{6-7}{2-1} = -1$. This means for every dollar increase in price, the company sells 1 million fewer watches. The equation of the line in the slope-intercept form is $x = mp + b$, where $m$ is the slope and $b$ is the y-intercept. Substituting the slope and one point into the equation, we get $7 = -1(1) + b$, solving for $b$ gives $b = 8$. Thus, the equation relating $x$ and $p$ is $x = -p + 8$.Substituting $x = -p + 8$ into the revenue equation $R = xp$, we get $R = p(-p + 8) = -p^2 + 8p$. To find the maximum revenue, we can complete the square or use the vertex formula for a parabola, $p = -\frac{b}{2a}$, where $a = -1$ and $b = 8$, giving $p = -\frac{8}{2(-1)} = 4$. Substituting $p = 4$ into the revenue equation, $R = -4^2 + 8(4) = -16 + 32 = 16$ million dollars. Therefore, the maximum revenue is $16,000,000 at $4 per watch.

A Company Wants to Know What Price They Should Charge