As part of the T-shirt fundraiser last year, the student council at Lincoln sold 920 T-shirts for four dollars each. This year the student council is thinking about selling T-shirts again. Based on a student survey for every dollar increase in T-shirt price there is a drop in sales by 140 T-shirts. What price should the student council sell each T-shirt if it needs to generate a revenue of $4970?

We can solve this problem by defining a function for revenue in terms of the price per T-shirt, then setting it equal to the desired revenue of $4,970 and solving for the price.

### Step 1: Define variables
- Let $ p $ be the price per T-shirt in dollars.
- Let $ q(p) $ be the number of T-shirts sold at price $ p $.

From the problem:
- Last year, the council sold 920 T-shirts for $4 each.
- For every $1 increase in price, 140 fewer T-shirts are sold.

So, the number of T-shirts sold at a price $ p $ can be written as:
\[
q(p) = 920 - 140(p - 4)
\]
This equation reflects that at $ p = 4 $, they sell 920 T-shirts, and for every dollar increase, the number sold decreases by 140.

### Step 2: Write the revenue equation
The revenue $ R $ is the product of the price per T-shirt and the number of T-shirts sold:
\[
R(p) = p \cdot q(p)
\]
Substitute the expression for $ q(p) $ into the revenue equation:
\[
R(p) = p \cdot (920 - 140(p - 4))
\]

### Step 3: Solve for the price when revenue is $4,970
Set the revenue equal to $4,970 and solve for $ p $:
\[
4970 = p \cdot (920 - 140(p - 4))
\]
First, simplify the expression inside the parentheses:
\[
4970 = p \cdot (920 - 140p + 560)
\]
\[
4970 = p \cdot (1480 - 140p)
\]
Expand the equation:
\[
4970 = 1480p - 140p^2
\]
Rearrange the equation to form a quadratic equation:
\[
140p^2 - 1480p + 4970 = 0
\]
Divide the whole equation by 10 to simplify:
\[
14p^2 - 148p + 497 = 0
\]

### Step 4: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[
p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where $ a = 14 $, $ b = -148 $, and $ c = 497 $.

Now let's compute the solution.It appears that the discriminant is negative, which means there are no real solutions to the quadratic equation. This suggests that it's impossible to achieve exactly $4,970 in revenue under the given conditions.

However, we can examine whether a nearby revenue value might be feasible or adjust the assumptions slightly. Would you like me to investigate other potential options for the price or clarify any details?

Here are five related questions for deeper exploration:
1. How would increasing the T-shirt price affect overall profits?
2. What if we wanted to achieve a revenue slightly less than $4,970?
3. Could there be alternative ways to raise the desired revenue aside from just adjusting the price?
4. How many T-shirts would be sold if the price were raised by $2?
5. What would happen if the price per T-shirt were reduced instead of increased?

**Tip:** When dealing with quadratic equations, always check the discriminant first to see if real solutions are possible.

Learn how to solve a revenue-based quadratic equation for a T-shirt fundraiser where sales decrease with increasing price. Explore algebraic methods and the quadratic formula to determine the optimal price for generating $4,970 in revenue. Suitable for students in Grades 9-11.

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