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
make a monthly profit of at least $3000
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.
To determine the range of prices that the charity could consider in order to make a monthly profit of at least $3000, we need to find the values of 𝑥 where the profit equation is greater than or equal to $3000.
The profit equation is given by 𝑦 = -35𝑥^2 + 1250𝑥 - 6500.
Setting 𝑦 ≥ $3000, we have:
-35𝑥^2 + 1250𝑥 - 6500 ≥ 3000
Rearranging the equation and simplifying:
-35𝑥^2 + 1250𝑥 - 9500 ≥ 0
To solve this quadratic inequality, we can use various methods, such as factoring or the quadratic formula. In this case, we will use the quadratic formula.
The quadratic formula is given by:
𝑥 = (-𝑏 ± √(𝑏^2 - 4𝑎𝑐)) / (2𝑎)
For our equation, 𝑎 = -35, 𝑏 = 1250, and 𝑐 = -9500.
Plugging these values into the quadratic formula:
𝑥 = (-(1250) ± √((1250)^2 - 4(-35)(-9500))) / (2(-35))
Simplifying further:
𝑥 = (-1250 ± √(1562500 - 1330000)) / (-70)
𝑥 = (-1250 ± √(232500)) / (-70)
𝑥 = (-1250 ± 482.93) / (-70)
We have two possible 𝑥 values:
1. 𝑥 = (-1250 + 482.93) / (-70) ≈ 19.1
2. 𝑥 = (-1250 - 482.93) / (-70) ≈ -27.1
Since the price of each T-shirt cannot be negative, the charity could consider prices between approximately $19.1 (rounded up) and higher in order to make a monthly profit of at least $3000.