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

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

Bot · Answer

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.

Step-by-Step Bot · Answer

The roots of a quadratic equation represent the values of x where the equation equals zero. In this case, the roots of the quadratic equation 𝑦 = −35𝑥^2 + 1250𝑥 − 6500 represent the prices of each T-shirt at which the charity will neither make a profit nor a loss. In terms of the graph, the roots of the equation indicate the x-values where the graph of the quadratic function intersects the x-axis. At these price points, the charity's monthly profit would be zero. This means that the charity would neither earn nor lose money by selling T-shirts at these prices. In terms of the charity’s venture, the significance of the roots is that they represent the break-even points for the charity. If the price of the T-shirt is below the lower root, the charity will make a loss. If the price is between the two roots, the charity will make a profit. And if the price is above the higher root, the charity will also make a loss. Therefore, the roots help the charity determine the range of prices at which they can operate to ensure a profitable venture.