Question

What is the best price at which they should sell the T-shirts to make the most money at their fundraiser?

Answer 1

I don’t have the details (demand/cost data) from your fundraiser, so I can’t give a numeric price. Tell me the price/quantity data or the demand/cost function and I’ll compute it. Meanwhile, here’s how to find the best price.

1) Clarify objective
- "Most money" usually means maximize revenue (price × quantity sold).
- If you mean maximize profit, you must include the cost per shirt (and fixed costs, if you care about total profit rather than per‑shirt).

2) If demand is approximately linear
Suppose quantity demanded q(p) = a − b p (a, b > 0).
- Maximize revenue R(p) = p·q(p) = p(a − b p).
dR/dp = a − 2b p = 0 → optimal revenue price p* = a/(2b).
- If there is constant unit cost c and you want to maximize profit Π(p) = (p − c)q(p):
Π(p) = (p − c)(a − b p). dΠ/dp = a − 2b p + b c = 0 → p* = (a + b c)/(2b) = a/(2b) + c/2.

3) If demand is some other function q(p)
- Write revenue R(p)=p q(p) (or profit Π(p)=(p−c) q(p)). Differentiate with respect to p and set derivative to 0, check second derivative is negative.

4) If you only have a couple of data points
- Fit a linear demand q(p) through the points to get a and b, then apply the formula above. If noisy data, do a regression.

If you provide: (a) price(s) vs quantity sold, or (b) an explicit demand function and the unit cost, I’ll compute the exact best price.