The daily profit, P, for selling x units of a certain item at a sporting goods store can be modeled...

Understanding the Question

We need to find the maximum daily profit for selling a certain item. The profit function is given as:

\(\mathrm{P(x)} = -\mathrm{a}(\mathrm{x} - \mathrm{b}/2\mathrm{a})^2 + \mathrm{b}^2/4\mathrm{a} + \mathrm{c}\)

where:

• a and b are positive constants
• c is a nonnegative constant
• x represents units sold

Key Insight - Recognizing Vertex Form

This profit function is already in vertex form - a huge time-saver! Since a > 0, the coefficient -a is negative, making this a downward-opening parabola.

For any parabola in vertex form \(\mathrm{P(x)} = -\mathrm{a}(\mathrm{x} - \mathrm{h})^2 + \mathrm{k}\):

• The vertex (maximum point) occurs at \(\mathrm{x} = \mathrm{h}\)
• The maximum value equals k

In our case:

• Maximum occurs at \(\mathrm{x} = \mathrm{b}/2\mathrm{a}\)
• Maximum profit = \(\mathrm{b}^2/4\mathrm{a} + \mathrm{c}\)

Therefore, to find the specific numerical value of the maximum profit, we need to determine \(\mathrm{b}^2/4\mathrm{a} + \mathrm{c}\).

Analyzing Statement 1

Statement 1: \(\mathrm{b}^2 + 4\mathrm{ac} = 52\mathrm{ac}/3\)

Let's see if this alone gives us a specific maximum profit value.

Starting with the given equation:

• \(\mathrm{b}^2 + 4\mathrm{ac} = 52\mathrm{ac}/3\)
• \(\mathrm{b}^2 = 52\mathrm{ac}/3 - 4\mathrm{ac}\)
• \(\mathrm{b}^2 = 52\mathrm{ac}/3 - 12\mathrm{ac}/3\)
• \(\mathrm{b}^2 = 40\mathrm{ac}/3\)

Now we can find \(\mathrm{b}^2/4\mathrm{a}\):

• \(\mathrm{b}^2/4\mathrm{a} = (40\mathrm{ac}/3) ÷ 4\mathrm{a} = 40\mathrm{c}/12 = 10\mathrm{c}/3\)

So the maximum profit becomes:

• Maximum profit = \(\mathrm{b}^2/4\mathrm{a} + \mathrm{c} = 10\mathrm{c}/3 + \mathrm{c} = 10\mathrm{c}/3 + 3\mathrm{c}/3 = \)\(13\mathrm{c}/3\)

This tells us the maximum profit equals \(13\mathrm{c}/3\), but without knowing the specific value of c, we cannot determine a numerical answer for the maximum profit.

Statement 1 alone is NOT sufficient. ❌

This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: \(\mathrm{c} = 360\)

With only this information:

• Maximum profit = \(\mathrm{b}^2/4\mathrm{a} + \mathrm{c} = \mathrm{b}^2/4\mathrm{a} + 360\)

We know \(\mathrm{c} = 360\), but we still don't know the values of a and b (and therefore cannot determine \(\mathrm{b}^2/4\mathrm{a}\)). Without this crucial piece, we cannot calculate the specific maximum profit value.

Statement 2 alone is NOT sufficient. ❌

This eliminates choice B.

Combining Both Statements

Now let's see what happens when we use both statements together.

From Statement 1, we derived: \(\mathrm{b}^2/4\mathrm{a} = 10\mathrm{c}/3\)
From Statement 2: \(\mathrm{c} = 360\)

Substituting \(\mathrm{c} = 360\) into our expression from Statement 1:

• \(\mathrm{b}^2/4\mathrm{a} = 10(360)/3 = 3600/3 = 1200\)

Therefore:

• Maximum profit = \(\mathrm{b}^2/4\mathrm{a} + \mathrm{c} = 1200 + 360 = \)1560

With both statements together, we can determine a unique, specific value for the maximum profit: $1560.

Both statements together ARE sufficient. ✓

[STOP - Sufficient!]

This eliminates choice E.

The Answer: C

Since neither statement alone is sufficient, but together they provide enough information to find the exact maximum profit, the answer is C.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."