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 larger slope for widgets means they are always more profitable than gadgets.
the fewer gadgets are made, the more profitable is each one.
the negative y - intercept for widgets means they will never be profitable.
fewer gadgets than widgets have to made for them to be profitable.

Explanation:

1. Find break-even point (profit = 0) for Widgets:

Set $15x - 300 = 0$
$15x = 300$
$x = 20$ (Need 20 Widgets to be profitable)

1. Find break-even point for Gadgets:

Set $3x - 30 = 0$
$3x = 30$
$x = 10$ (Need 10 Gadgets to be profitable)

1. Evaluate other options:

• Widgets are not always more profitable: at $x=15$, Widget profit = $15(15)-300=-75$, Gadget profit = $3(15)-30=15$.
• Gadgets have a positive slope, so more made = more profit per unit, not fewer.
• Widgets become profitable once 20 are made, so they can be profitable.

1. Only the final statement is true: 10 (Gadgets) < 20 (Widgets) for profitability.

Answer:

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