14. 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.

Question

a) Determine the roots of the equation

Answer 1

To find the roots of the equation, we need to find the values of 𝑥 that make 𝑦 equal to zero. So we set 𝑦 equal to zero and solve for 𝑥:

0 = −35𝑥2 + 1250𝑥 − 6500

We can simplify this equation by dividing both sides by 5:

0 = −7𝑥2 + 250𝑥 − 1300

Now we can use the quadratic formula to find the roots:

𝑥 = (−𝑏 ± √𝑏^2 − 4𝑎𝑐) / (2𝑎)

Where 𝑎 = −7, 𝑏 = 250, and 𝑐 = −1300. Plugging these values into the formula, we get:

𝑥 = (−250 ± √250^2 − 4(−7)(−1300)) / (2(−7))

Simplifying this expression, we get:

𝑥 = (−250 ± √237500) / (−14)

Now we can use a calculator to approximate the roots:

𝑥 ≈ 43.7 or 𝑥 ≈ 32.1

So the roots of the equation are approximately 43.7 and 32.1. These values represent the prices of the T-shirts that will result in zero profit for the charity. Above these prices, the charity will make a profit, and below these prices, the charity will incur a loss.