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

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.

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.

Answer 2

To determine the roots of the equation, we need to find the values of x where y equals zero.

The equation is given as y = -35x^2 + 1250x - 6500.

Setting y = 0, we have:

0 = -35x^2 + 1250x - 6500.

Now, we need to solve this quadratic equation for x. We can do this by factoring or by using the quadratic formula.

Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a),

where a, b, and c are the coefficients from the quadratic equation, y = -35x^2 + 1250x - 6500.

In this case, a = -35, b = 1250, and c = -6500.

Plugging these values into the quadratic formula, we get:

x = (-1250 ± √(1250^2 - 4(-35)(-6500))) / (2(-35)).

Simplifying further:

x = (-1250 ± √(1562500 - 910000)) / (-70).

x = (-1250 ± √(652500)) / (-70).

x = (-1250 ± √(900 * 725)) / (-70).

x = (-1250 ± √900 * √725) / (-70).

x = (-1250 ± 30√725) / (-70).

So, the two roots of the equation are:

x1 = (-1250 + 30√725) / (-70).

x2 = (-1250 - 30√725) / (-70).

Therefore, the roots of the equation are x1 = (-1250 + 30√725) / (-70) and x2 = (-1250 - 30√725) / (-70).