b The manufacturer must sell between 30 000 and 98 000 digital audio players to earn a profit of at least $100 000 in a month.
C
cAudio players the manufacturer needs to sell: between 47 000 and 81 000 digital audio players to earn a profit of at least $100 000 in a month. Explanation: See solution.
Practice makes perfect
a We are given a model for the profit of an electronics company.
P(x)=x(-27.5x+3520)+20 000
Let's write and simplify an inequality representing that the profit is at least $100 000.
We can use calculator to sketch the graph of the quadratic corresponding to this inequality.
We begin by pushing the Y= button and typing the equation in the first row.
To see the graph you will need to adjust the window.
Push WINDOW, change the settings, and push GRAPH.
Let's copy the graph from the calculator screen. Since x represents a selling quantity, only positive values are meaningful. The solution of the inequality is the set of values of x for which the graph is not below the horizontal axis.
b From the graph of Part A, we see that the selling quantities to earn at least $100 000 profit are between two bounds. To find these bounds we either use a calculator or solve a quadratic equation.
-27.5x^2+3520x-80 000=0
Let's follow the graphical approach and use a calculator to find the x-intercepts of the graph.
From the screen with your graph, push 2nd and TRACE and choose zero from the menu. The calculator will prompt you to choose a left and right bound and to provide the calculator with a best guess of where the zero might be.
The results we obtained give bounds for the possible values of x satisfying the inequality of Part A.
29.55≤ x≤ 98.45
If we round these bounds to the nearest integer, this means that the manufacturer must sell between 30 and 98 thousand digital audio players to earn a profit of at least $100 000 in a month.
c If the manufacturer has an additional expense of $25 000, then the profit decreases by the same amount.
P(x)=x(-27.5x+3520)+20 000-25 000
⇕
P(x)=x(-27.5x+3520)-5000
This subtraction represents a translation down 25 000 units of the graph of the profit function. This shift also applies to the inequality we need to solve when looking for quantities to sell to reach $100 000 profit.
-27.5x^2+3520x-80 000-25 000≥ 0
⇕
-27.5x^2+3520x-105 000≥ 0
This reduces the interval where the manufacturer can reach at least $100 000 profit.
We can use the calculator again to find the x-intercepts of the new graph.
The results we obtained give bounds for the possible values of x satisfying the new inequality.
47.33≤ x≤ 80.67
If we round these bounds to the nearest integer, with the increased expenses the manufacturer must sell between 47 and 81 thousand digital audio players to earn a profit of at least $100 000 in a month.
