The amount of money that a local charity earns by selling T-shirts at a mall depends on the price of each T-shirt. The monthly profit, 𝑦, in dollars is given by the quadratic equation 𝑦 = −35𝑥2 + 1250𝑥 − 6500 where 𝑥 represents the price of each T-shirt Under what circumstances will the charity maximize its profit (make the most money)?
The charity will maximize its profit when it reaches the vertex of the quadratic equation.
The x-coordinate of the vertex is given by:
𝑥 = −𝑏/2𝑎
where 𝑏 = 1250 and 𝑎 = −35.
Substituting these values and simplifying:
𝑥 = −1250/(2(−35))
𝑥 = 17.86
Therefore, the charity will maximize its profit when they sell T-shirts for $17.86 each.
To find the circumstances under which the charity will maximize its profit, we need to determine the vertex of the quadratic equation 𝑦 = −35𝑥^2 + 1250𝑥 − 6500. The vertex represents the maximum point on the graph of the quadratic equation.
The vertex of a quadratic equation in the form 𝑦 = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 can be found using the formula 𝑥 = -𝑏 / (2𝑎).
In this case, 𝑎 = -35 and 𝑏 = 1250. Plugging these values into the formula, we have:
𝑥 = -1250 / (2(-35))
𝑥 = -1250 / (-70)
𝑥 = 17.857
Therefore, when the price of each T-shirt is $17.857, the charity will maximize its profit and make the most money.