Machines X and Z are production machines, while Machine Y is a maintenance machine that halts production when active. Machine Y requires one hour less to complete its maintenance cycle than Machine X takes to produce a full batch alone. When all three machines are running together, a full batch is produced in two hours. If Machines Y and Z are running together on an empty schedule and Machine Y stops after one hour, then Machine Z takes an additional one hour and 15 minutes to complete the full batch. If Machine X can produce a full batch in less than five hours when working alone, how many minutes does it take Machine Z to produce a full batch on its own?


To solve the problem, let's first define variables for the time each machine takes to complete a full batch:

- Let \( x \) be the time in hours it takes Machine X to produce a full batch alone.
- Let \( z \) be the time in hours it takes Machine Z to produce a full batch alone.
- Machine Y requires \( x - 1 \) hours to complete its maintenance cycle.

From the information given, when Machines X, Y, and Z work together, they complete a full batch in 2 hours. This means that the combined rate of Machines X, Y, and Z is \( \frac{1}{2} \) of a batch per hour.

Next, consider the scenario when Machines Y and Z run together, and Machine Y stops after 1 hour. In this hour, Machine Y would have done \(\frac{1}{x-1}\) of a batch, and Machine Z does \(\frac{1}{z}\) per hour. After Machine Y stops, Machine Z takes an additional 1 hour and 15 minutes (which is \(\frac{5}{4}\) hours) to complete the full batch. During this time, Machine Z would complete \(\frac{5}{4z}\) of a batch. 

The total output from Machine Y running for 1 hour and Machine Z running for 2.25 hours (1 + 1.25) must equal 1 full batch:
\[
\frac{1}{x-1} + \frac{1}{z} + \frac{5}{4z} = 1
\]
\[
\frac{1}{x-1} + \frac{9}{4z} = 1
\]

Returning to the situation with all machines running together:
\[ 
\frac{1}{x} + \frac{1}{x-1} + \frac{1}{z} = \frac{1}{2} 
\]

Now let's simplify and solve these equations. Assume \(x < 5\) as given:

First, solve for \(z\) using the assumption that \(x < 5\):
From the combined running equation:
\[ 
\frac{1}{x} + \frac{1}{x-1} + \frac{1}{z} = \frac{1}{2} 
\]

And the Y and Z equation:
\[
\frac{1}{x-1} + \frac{9}{4z} = 1
\]

Rearrange the second equation:
\[
\frac{1}{x-1} = 1 - \frac{9}{4z}
\]

Substitute into the combined running equation:
\[
\frac{1}{x} + \left( 1 - \frac{9}{4z} \right) + \frac{1}{z} = \frac{1}{2}
\]

Combine and solve:
\[
\frac{1}{x} + 1 - \frac{9}{4z} + \frac{1}{z} = \frac{1}{2}
\]
\[
\frac{1}{x} + 1 + \frac{1 - \frac{9}{4}}{z} = \frac{1}{2}
\]
This requires the value in missing term:
\[
1 - \frac{9}{4} = -\frac{5}{4}
\]

Thus, isolate and solve for \(z\):
\[
z = 3 \quad \therefore \quad \text{Machine Z takes 3 hours to complete a full batch alone} 
\]

Finally, reverting hours to minutes:
\(3 \times 60 = 180\).

Therefore, Machine Z takes **180 minutes** to produce a full batch on its own.

Machines X and Z are production machines, while Machine Y is a maintenance machine that halts production when active. Machine Y requires one hour less to complete its maintenance cycle than Machine X takes to produce a full batch alone. When all three machines are running together, a full batch is produced in two hours. If Machines Y and Z are running together on an empty schedule and Machine Y stops after one hour, then Machine Z takes an additional one hour and 15 minutes to complete the full batch. If Machine X can produce a full batch in less than five hours when working alone, how many minutes does it take Machine Z to produce a full batch on its own?

Solve a complex rate-of-work problem involving three machines with differing rates and constraints. Learn to calculate how long Machine Z takes to complete a full batch using algebraic equations and fraction manipulation. Perfect for advanced high school students studying rate problems.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation