https://file.aiquickdraw.com/mathbot/file/17301553141775cul2p1v.jpg

To solve this problem, let's define the variables and set up an equation based on the information provided.

1. **Define Variables:**
   - Let \( s \) represent the time (in minutes) it takes Machine A to fill a case by itself.
   - Machine B takes \( s + 10 \) minutes to fill a case by itself.

2. **Determine Rates:**
   - Machine A's filling rate is \( \frac{1}{s} \) cases per minute.
   - Machine B's filling rate is \( \frac{1}{s + 10} \) cases per minute.

3. **Combined Rate:**
   - When both machines work together, their combined rate is the sum of their individual rates:
     \[
     \frac{1}{s} + \frac{1}{s + 10}
     \]
   - Together, they take 15 minutes to fill one case, so their combined rate is also \( \frac{1}{15} \) cases per minute.

4. **Set Up the Equation:**
   \[
   \frac{1}{s} + \frac{1}{s + 10} = \frac{1}{15}
   \]

5. **Solve the Equation:**
   Multiply both sides by \( 15s(s + 10) \) to eliminate the denominators:
   \[
   15(s + 10) + 15s = s(s + 10)
   \]
   Expanding and simplifying this equation will allow us to solve for \( s \).

Let me solve this equation for you.The solutions to the equation are approximately:

- \( s \approx -5.81 \) (which we discard since time cannot be negative)
- \( s \approx 25.81 \)

Thus, **Machine A takes approximately 25.81 minutes to fill a case**, and **Machine B takes \( 25.81 + 10 = 35.81 \) minutes** to fill a case.

Would you like further details on this solution or have any other questions?

Here are some related questions for practice:
1. What would happen to the combined filling time if Machine A's rate doubled?
2. How would the equation change if Machine B took \( s + 5 \) minutes instead?
3. If Machine A were twice as fast, how would that affect the time taken to fill the case together?
4. How long would it take if a third machine, Machine C, filled cases in \( s + 15 \) minutes and worked alongside A and B?
5. What if the two machines had to fill two cases together; how long would that take?

**Tip:** When dealing with combined rates, add individual rates to find the combined rate.

The Turtledove Chocolate factory has two chocolate machines. Machine A takes s minutes to fill a case with chocolates, and machine B takes s + 10 minutes to fill a case. Working together, the two machines take 15 min to fill a case. Approximately how long does each machine take to fill a case?

Learn how to solve a combined rates problem involving two machines at the Turtledove Chocolate factory. This problem involves setting up and solving equations for rates and work, suitable for high school students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation