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 Determine the range of prices that the charity could consider, in order to

Question

make a monthly profit of at least $3000

Answer 1

To find the range of prices that will result in a monthly profit of at least $3000, we need to solve the inequality:

-35𝑥^2 + 1250𝑥 − 6500 ≥ 3000

First, we can simplify the inequality by subtracting 3000 from both sides:

-35𝑥^2 + 1250𝑥 − 9500 ≥ 0

Next, we can factor out -5 from the polynomial:

-5(7𝑥^2 - 250𝑥 + 1900) ≥ 0

Now we can solve for the values of 𝑥 that make the inequality true by analyzing the sign of each factor.

The quadratic factor inside the parentheses can be factored using the quadratic formula:

𝑥 = (250 ± sqrt(250^2 - 4(7)(1900))) / (2(7))
𝑥 = 31.67 or 5.68

These are the two values of 𝑥 where the quadratic factor equals zero. We can use these values to divide the number line into three intervals:

(−∞, 5.68], [5.68, 31.67], [31.67, ∞)

Next, we can test a point in each interval to determine the sign of the factor. For example, we can choose 𝑥 = 0 for the interval (−∞, 5.68], 𝑥 = 10 for the interval [5.68, 31.67], and 𝑥 = 40 for the interval [31.67, ∞]. Plugging these values into the factor, we get:

-5(7(0)^2 - 250(0) + 1900) = 9500 > 0
-5(7(10)^2 - 250(10) + 1900) = -31500 < 0
-5(7(40)^2 - 250(40) + 1900) = 63000 > 0

Therefore, the inequality is true for 𝑥 in the intervals (−∞, 5.68] and [31.67, ∞]. These correspond to the price ranges:

𝑥 ≤ 5.68 or 𝑥 ≥ 31.67

So, the local charity could consider selling T-shirts for prices that are less than or equal to $5.68 or greater than or equal to $31.67 in order to make a monthly profit of at least $3000.