You want to start an on-campus business making and selling custom-made t-shirts.  After some research,  you estimate that the total cost of making T t-shirts is given by the function

C left parenthesis T right parenthesis equals 6.5 T plus 600

dollars.  When you set a unit price of p dollars per t-shirt, there will be a demand of T t-shirts, given by the demand equation

75 p plus 2 T equals 3000.

Given this information, what unit price will maximize revenue?

Round your answer appropriately and input just the number that you get.  Do not include any labels or units. 



To solve for the unit price \( p \) that maximizes revenue, we follow these steps:

### Step 1: Express \( T \) in terms of \( p \)
We are given the demand equation:
\[
75p + 2T = 3000
\]
We can solve for \( T \) in terms of \( p \):
\[
2T = 3000 - 75p
\]
\[
T = \frac{3000 - 75p}{2}
\]

### Step 2: Define the revenue function
Revenue \( R \) is the total income from selling \( T \) t-shirts at price \( p \), which is given by:
\[
R = p \times T
\]
Substitute \( T \) from the demand equation:
\[
R = p \times \frac{3000 - 75p}{2}
\]
\[
R = \frac{p(3000 - 75p)}{2}
\]
\[
R = \frac{3000p - 75p^2}{2}
\]
\[
R = 1500p - 37.5p^2
\]

### Step 3: Maximize the revenue function
To find the price \( p \) that maximizes revenue, take the derivative of \( R \) with respect to \( p \) and set it equal to zero:
\[
\frac{dR}{dp} = 1500 - 75p
\]
Set the derivative equal to zero to find the critical points:
\[
1500 - 75p = 0
\]
\[
75p = 1500
\]
\[
p = \frac{1500}{75} = 20
\]

Thus, the unit price that maximizes revenue is \( \boxed{20} \). 

Would you like further details, or do you have any questions?

Here are 5 related questions:
1. How would the maximum revenue change if the fixed cost was different?
2. What is the total revenue at the price \( p = 20 \)?
3. How would an increase in production cost per t-shirt affect the price to maximize revenue?
4. Can the demand equation change over time, and how would that affect the price \( p \)?
5. How would you optimize production to minimize costs at different demand levels?

**Tip**: In optimization problems, always consider taking derivatives to find maximum or minimum values efficiently!

You want to start an on-campus business making and selling custom-made t-shirts. After some research, you estimate that the total cost of making T t-shirts is given by the function C(T) = 6.5T + 600 dollars. When you set a unit price of p dollars per t-shirt, there will be a demand of T t-shirts, given by the demand equation 75p + 2T = 3000. Given this information, what unit price will maximize revenue? Round your answer appropriately and input just the number that you get. Do not include any labels or units.

Discover how to maximize revenue for a custom t-shirt business by analyzing the relationship between price, demand, and revenue. This problem uses algebraic concepts, including demand and revenue equations, and explores optimization techniques such as taking the derivative of a quadratic function.

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