a company plans to sell embroidered hats for $15. the company's financial planner estimates that the cost, y, of manufacturing the hats is a quadratic function with a y-intercept of 7,920 and a vertex of (150, 9,000). what is the minimum number of hats the company must sell to make a profit?
To calculate the minimum number of hats the company must sell to make a profit, we need to determine the break-even point, where the cost of manufacturing the hats is equal to the revenue generated from selling them.
The cost, y, of manufacturing the hats is given as a quadratic function with a y-intercept of 7,920 and a vertex of (150, 9,000). This means that the cost can be represented as a quadratic equation in the form: y = ax^2 + bx + c, where x is the number of hats sold.
We are given the vertex of the quadratic function, which corresponds to the break-even point (the quantity of hats that result in zero profit or loss). The x-coordinate of the vertex (150) represents the number of hats at the break-even point. So at this point, the revenue generated from selling x hats is equal to the cost of manufacturing those hats.
Let's find the actual equation of the quadratic function representing the cost of manufacturing the hats using the given information.
First, we know that the y-intercept is 7,920. This means that when no hats are sold (x = 0), the cost is 7,920. So we have the point (0, 7,920).
Next, we know the vertex is (150, 9,000). The vertex form of a quadratic function is given by: y = a(x-h)^2 + k, where (h, k) represents the vertex. In this case, h = 150 and k = 9,000. Substituting these values into the vertex form equation, we get: y = a(x-150)^2 + 9,000.
Now we can substitute the coordinates of one of the given points (either the y-intercept or the vertex) into the equation and solve for a.
Using the y-intercept (0, 7,920):
7,920 = a(0-150)^2 + 9,000
7,920 = a(22500) + 9,000
7,920 - 9,000 = 22,500a
-1,080 = 22,500a
a = -1,080 / 22,500
a = -0.048
Now we have the equation for the cost of manufacturing the hats: y = -0.048(x-150)^2 + 9,000
To find the break-even point, we set the cost equal to the revenue and solve for x:
-0.048(x-150)^2 + 9,000 = 15x
Simplifying the equation, we get:
-0.048x^2 + 14.4x + 9,000 = 15x
Rearranging the terms:
-0.048x^2 - 0.6x + 9,000 = 0
We can now use the quadratic formula to find the solutions for x, which represent the quantity of hats sold. The formula is:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -0.048, b = -0.6, and c = 9,000. Substituting these values into the quadratic formula:
x = (-(-0.6) ± √((-0.6)^2 - 4(-0.048)(9,000))) / (2(-0.048))
Simplifying further:
x = (0.6 ± √(0.36 + 1728)) / (-0.096)
x = (0.6 ± √1728.36) / (-0.096)
Calculating √1728.36 ≈ 41.6, we have:
x ≈ (0.6 ± 41.6) / (-0.096)
For simplicity, let's calculate both the positive (+) and negative (-) solutions:
For the positive solution:
x ≈ (0.6 + 41.6) / (-0.096)
x ≈ 42.2 / (-0.096)
x ≈ -439.583...
For the negative solution:
x ≈ (0.6 - 41.6) / (-0.096)
x ≈ -41 / (-0.096)
x ≈ 427.083...
Since the number of hats sold cannot be negative, we conclude that the minimum number of hats the company must sell to make a profit is approximately 428 hats.
To find the minimum number of hats the company must sell to make a profit, we need to determine the point at which the revenue (sales) exceeds the cost.
Given that the hats are sold for $15 each, the revenue generated by selling "x" hats can be expressed as:
Revenue = 15x
Now, we can find the cost function using the given information. Since the vertex of the quadratic function is (150, 9,000), and the y-intercept is 7,920, we can write the cost function in vertex form as:
y = a(x - h)^2 + k
Where (h, k) represents the vertex. In this case, we have:
y = a(x - 150)^2 + 9,000
Using the y-intercept, we can substitute the coordinates (0, 7,920) into the equation to find the value of "a" (the coefficient of the quadratic function):
7,920 = a(0 - 150)^2 + 9,000
7,920 = 22,500a - 9,000
16,920 = 22,500a
a = 16,920 / 22,500
a = 0.75
Thus, the cost function is given by:
y = 0.75(x - 150)^2 + 9,000
Now, to find the break-even point or the minimum number of hats to make a profit, we need to set the cost equal to the revenue:
15x = 0.75(x - 150)^2 + 9,000
Simplifying the equation:
15x = 0.75x^2 - 225x + 16875 + 9,000
0.75x^2 - 240.75x + 26,875 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Since the vertex is given, we can use the factored form:
0.75(x - 150)(x - 150) = 0
Simplifying:
(x - 150)(x - 150) = 0
Setting each factor equal to zero:
x - 150 = 0
x = 150
Therefore, to make a profit, the company needs to sell more than 150 hats, specifically 151 or more hats.
revenue for x hats is y=15x
cost is y=0.4(x-150)^2+9000
So, you want x when
15x >= 0.4(x-150)^2+9000
x >= 262.5
so, at least 263 hats