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The given question asks us to find expressions related to bicycle sales and revenue.

### Step 1: Given Information
- The price function per bicycle \( p(x) \) is given by:
\[
p(x) = 150 - \frac{x}{4}
\]
where \( x \) is the number of bicycles sold, and \( p(x) \) is the price per bicycle.

### Part a: Expression for Total Revenue
Total revenue \( R(x) \) is found by multiplying the price per bicycle by the number of bicycles sold, i.e.,
\[
R(x) = p(x) \cdot x
\]
Substitute the expression for \( p(x) \) into the equation:
\[
R(x) = \left( 150 - \frac{x}{4} \right) x
\]
Simplify this:
\[
R(x) = 150x - \frac{x^2}{4}
\]
Thus, the total revenue function is:
\[
R(x) = 150x - \frac{x^2}{4}
\]

### Part b: Maximizing Revenue
To find the number of bicycle sales that leads to maximum revenue, we need to maximize the revenue function \( R(x) \). This involves finding the critical points by taking the derivative of \( R(x) \) and setting it equal to zero.

First, take the derivative of \( R(x) \):
\[
R'(x) = 150 - \frac{2x}{4}
\]
Simplify:
\[
R'(x) = 150 - \frac{x}{2}
\]
Now, set \( R'(x) = 0 \) to find the critical points:
\[
150 - \frac{x}{2} = 0
\]
\[
\frac{x}{2} = 150
\]
\[
x = 300
\]

### Conclusion
- The total revenue function is \( R(x) = 150x - \frac{x^2}{4} \).
- The number of bicycle sales that leads to maximum revenue is \( x = 300 \).

Would you like more details or have any other questions?

### Follow-up Questions:
1. How would the revenue function change if the price formula had a different constant?
2. What is the maximum possible revenue with \( x = 300 \)?
3. How does the revenue change as the number of bicycles sold increases beyond 300?
4. Can you verify the concavity of the function at \( x = 300 \) to confirm it's a maximum?
5. What happens to the price per bicycle when 300 units are sold?

### Tip:
When finding maximum or minimum points for revenue or profit functions, always check the second derivative to confirm if it’s a maximum or minimum!

The manager of a bicycle shop has found that, at a price (in dollars) of p(x) = 150 - x/4 per bicycle, x bicycles will be sold. a) Find an expression for the total revenue from the sale of x bicycles. b) Find the number of bicycle sales that leads to maximum revenue.

Learn how to find the revenue function and determine the maximum revenue in a bicycle sales problem. This involves differentiating a quadratic revenue function and solving for the critical points. Suitable for students in Grades 10-12.

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