Question

for a certain company, the equation used to determine profit y on x number of widgets is y = 15x - 300. similarly, the equation for profit on x number of gadgets is y = 3x - 30. if the equations were graphed together, which of these statements is true? the negative y - intercept for widgets means they will never be profitable. the fewer gadgets are made, the more profitable is each one. fewer gadgets than widgets have to made for them to be profitable. the larger slope for widgets means they are always more profitable than gadgets.

Explanation:

Step1: Recall the break - even point concept

The break - even point is when profit $y = 0$. For Widgets, set $y=15x - 300=0$, then $15x=300$, and $x = 20$. For Gadgets, set $y = 3x-30=0$, then $3x=30$, and $x = 10$.

Step2: Analyze each option

• Option 1: A negative $y$-intercept just means there is an initial loss. When $x$ is large enough, the profit can be positive. So Widgets can be profitable.
• Option 2: The profit per unit is the slope. For Gadgets, the profit per unit is 3. Making fewer gadgets does not change the profit per unit.
• Option 3: Since the break - even point for Gadgets is $x = 10$ and for Widgets is $x=20$, fewer Gadgets have to be made for them to be profitable.
• Option 4: The slope indicates the rate of change of profit with respect to the number of units. But it does not mean Widgets are always more profitable. For small values of $x$, Gadgets can be more profitable.

Answer:

C. Fewer Gadgets than Widgets have to be made for them to be profitable.