To solve this problem, we need to analyze the requirements for each studio's specialized art class profit goals and determine how many people, $p$, need to attend each class based on the pricing structure.

Information Summary:

1. Studio Costs and Profit Goals:

   • Art Studio A:
     • Expenses: $100 per class.
     • Profit goal: At least $400 per class.
   • Art Studio B:
     • Expenses: $200 per class.
     • Profit goal: At least $800 per class.

2. Group Class Rate:

   • Both studios charge $d$ dollars per person for the group class.

Analysis for Each Studio:

1. Art Studio A Requirement:

   • Total revenue goal = Expenses + Profit goal = $100 + 400 = 500$ dollars.
   • Since the rate per person is $d$, the total revenue generated with $p$ people is $p \cdot d$.
   • Set up the inequality for Art Studio A: $p \cdot d \geq 500$
   • Solving for $p$, we get: $p \geq \frac{500}{d}$

2. Art Studio B Requirement:

   • Total revenue goal = Expenses + Profit goal = $200 + 800 = 1000$ dollars.
   • For Art Studio B, the revenue with $p$ people at $d$ dollars per person must satisfy: $p \cdot d \geq 1000$
   • Solving for $p$, we get: $p \geq \frac{1000}{d}$

Answer Choices Analysis:

Now let's match these inequalities to the given answer choices:

• A: $p < \frac{1}{1000}d$ — Incorrect, as it does not match either of our inequalities.
• B: $p > \frac{1000}{d}$ — Incorrect, as it suggests $p$ needs to be greater than $\frac{1000}{d}$, rather than at least equal to it.
• C: $p < \frac{1}{d}$ — Incorrect, unrelated to the inequalities we derived.
• D: For Art Studio A, the number of people that must attend is at least the quotient of $500$ and $d$. Correct because $p \geq \frac{500}{d}$ is what we found for Art Studio A.
• E: For Art Studio B, the number of people that must attend is $p > \frac{1000}{d}$. — Incorrect, as it should be $p \geq \frac{1000}{d}$.
• F: For Art Studio B, the number of people that must attend is at least the quotient of $1000$ and $d$. Correct because $p \geq \frac{1000}{d}$ matches the requirement for Art Studio B.

Conclusion:

The correct answers are:

• D: For Art Studio A, $p$ must be at least $\frac{500}{d}$.
• F: For Art Studio B, $p$ must be at least $\frac{1000}{d}$.

Would you like more detailed steps or have any further questions?

Related Questions:

1. What would the inequalities be if the profit goals changed to a different value?
2. How would the solution change if the studios had different rates for each person attending?
3. What are some real-world applications of solving inequalities like these?
4. How would the answer choices change if the expenses were split among multiple classes?
5. Can this approach be used to solve revenue goals in other types of businesses?

Tip:

When solving for inequalities, always verify each part of the problem to ensure the solution aligns precisely with the required conditions.