Question

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 e) Describe the significance of the roots in terms of the graph and in terms of the charity’s venture

Answer 1

The roots of the quadratic equation represent the values of 𝑥 where the monthly profit, 𝑦, is equal to zero. In terms of the graph, the roots represent the x-intercepts, which are the points where the graph intersects the x-axis. In terms of the charity's venture, the roots represent the prices of the T-shirts where the charity neither makes a profit nor incurs a loss. This means that the charity should avoid pricing their T-shirts below or above these values in order to maximize their profit. Additionally, if the charity wants to determine the range of prices that will result in a profit, they can use the discriminant of the quadratic equation to determine the nature of the roots. If the discriminant is positive, there are two distinct roots, which means that there is a range of prices that will result in a profit. If the discriminant is zero, there is only one root, which means that the charity will break even at that price. Finally, if the discriminant is negative, there are no real roots, which means that the charity will incur a loss at all prices.