The weekly cost C of producing x units in a manufacturing process is given by C(x)=60x+2250. The number of units x produced in t hours is given by x(t)=50t. Find and interpret (*C* ∘ *x*)(t).

The composition \( (C \circ x)(t) \) represents the cost \( C \) as a function of time \( t \). This means you are looking for the weekly cost as a function of the hours \( t \) spent in production.

To find \( (C \circ x)(t) \), we substitute \( x(t) = 50t \) into the cost function \( C(x) \).

Given:
- \( C(x) = 60x + 2250 \)
- \( x(t) = 50t \)

Now substitute \( x(t) \) into \( C(x) \):

\[
(C \circ x)(t) = C(x(t)) = 60(50t) + 2250
\]

Simplifying:

\[
(C \circ x)(t) = 3000t + 2250
\]

### Interpretation:
The function \( (C \circ x)(t) = 3000t + 2250 \) represents the total weekly cost as a function of the number of hours \( t \) spent producing goods. 

- The term \( 3000t \) shows that the cost increases by 3000 for each additional hour of production.
- The constant 2250 represents the fixed weekly cost, regardless of how many hours are spent in production.

Would you like a deeper breakdown of these steps?

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Here are 5 related questions you might explore:
1. How would the function change if the fixed cost was higher?
2. What would happen if the production rate \( x(t) \) was different?
3. How does this cost function change if production runs for fewer or more hours?
4. How would you find the time \( t \) required for a specific cost?
5. What is the cost after 10 hours of production?

**Tip:** Always verify that units in your functions align when interpreting real-world relationships (e.g., cost per hour or cost per unit).

The weekly cost C of producing x units in a manufacturing process is given by C(x)=60x+2250. The number of units x produced in t hours is given by x(t)=50t. Find and interpret (C ∘ x)(t).

Explore the function composition problem where the weekly cost of producing goods is modeled by C(x) = 60x + 2250 and the number of units produced in t hours is given by x(t) = 50t. Learn how to find and interpret (C ∘ x)(t), a key concept in algebra. Suitable for high school students in Grades 10-12.

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