解题思路---**Question Stem:**
A local hardware store's profit P (measured in dollars) earned by selling indoor grills is given by the equation P = 5g² - 1,125, where g is the number of grills sold. How many grills must be sold to break even (that is, earn zero dollars in profit)?
**Options:**
a. 5
b. 15
c. 50
d. 225
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To understand the break-even point, we need to know that it's when a business earns zero profit. For this hardware store selling indoor grills, the profit is given by P equals 5g squared minus 1125, where g is the number of grills sold. To find the break-even point, we set the profit equal to zero, giving us the equation: 0 equals 5g squared minus 1125.
Now let's solve the equation step by step. Starting with 0 equals 5g squared minus 1125, we first add 1125 to both sides to get 1125 equals 5g squared. Next, we divide both sides by 5 to get 225 equals g squared. Finally, we take the square root of both sides to get g equals plus or minus 15.
From our solution g equals plus or minus 15, we only consider the positive solution g equals 15, since selling negative grills doesn't make business sense. Let's verify this answer by substituting g equals 15 back into the original equation. P equals 5 times 15 squared minus 1125, which equals 5 times 225 minus 1125, which equals 1125 minus 1125, which equals zero. This confirms that 15 grills is indeed the break-even point.
Let's solve this break-even analysis problem. A hardware store's profit from selling indoor grills is given by the equation P equals 5g squared minus 1,125, where g is the number of grills sold. We need to find how many grills must be sold to break even, meaning to earn zero dollars in profit.
To find the break-even point, we set the profit P equal to zero. Starting with P equals 5g squared minus 1,125, we substitute zero for P. This gives us 0 equals 5g squared minus 1,125. Adding 1,125 to both sides, we get 5g squared equals 1,125. Dividing both sides by 5, we find g squared equals 225. Taking the square root of both sides, g equals 15.
Let's verify our answer by substituting g equals 15 back into the original equation. P equals 5 times 15 squared minus 1,125. This equals 5 times 225 minus 1,125, which equals 1,125 minus 1,125, giving us P equals 0. This confirms our answer is correct. Therefore, 15 grills must be sold to break even, making the answer choice b.
Let's visualize the profit function graphically. The equation P equals 5g squared minus 1125 forms a parabola that opens upward. The break-even point occurs at coordinates 15, 0 where the parabola crosses the x-axis. Above the x-axis is the profit region, while below is the loss region. The vertex of the parabola at coordinates 0, negative 1125 shows the maximum loss occurs when no grills are sold.
Let's systematically check each answer choice by substituting into the original equation. For option a, g equals 5: P equals 5 times 25 minus 1125, which equals negative 1000, not zero. For option b, g equals 15: P equals 5 times 225 minus 1125, which equals 1125 minus 1125, which equals zero. This is correct! For option c, g equals 50: P equals 5 times 2500 minus 1125, which equals 11,375, not zero. For option d, g equals 225: P equals 5 times 50,625 minus 1125, which equals 252,000, not zero. Therefore, the answer is b, 15 grills.