The graph of the function C(x) = −0.34x2 + 12x + 62 is shown. The function models the production cost, C, in thousands of dollars for a tire company to manufacture a tire, where x is the number of tires produced, in thousands:
graph of a parabola opening down passing through points negative 4 and 57 hundredths comma zero, zero comma 62, 1 and 12 hundredths comma 75, 17 and 65 hundredths comma 167 and 55 hundredths, 34 and 18 hundredths comma 75, and 39 and 87 hundredths comma zero
If the company wants to keep its production costs under $75,000, then which constraint is reasonable for the model?
C(x) < 75, where C(x) is the cost in thousands of dollars and x is the number of tires produced in thousands.
To determine which constraint is reasonable for the model, we need to consider the maximum production cost the company wants to stay under. In this case, the maximum production cost is $75,000.
We can find the maximum cost by observing the graph of the function C(x) = -0.34x^2 + 12x + 62. Since the graph is a parabola opening downwards, the maximum cost will occur at the vertex of the parabola.
To find the x-coordinate of the vertex, we can use the formula x = -b / (2a) from the vertex form of a quadratic equation (y = ax^2 + bx + c).
In this case, a = -0.34 and b = 12.
x = -12 / (2 * -0.34)
x = -12 / -0.68
x = 17.65 (rounded to two decimal places)
The x-coordinate of the vertex is 17.65, which represents the number of thousands of tires produced. To find the cost associated with this production level, we can substitute x = 17.65 into the equation C(x).
C(17.65) = -0.34(17.65)^2 + 12(17.65) + 62
C(17.65) ≈ 167.55 (rounded to two decimal places)
The cost at the vertex is approximately $167,550. Since this is greater than $75,000, it means the company cannot keep its production cost under $75,000.
Therefore, the reasonable constraint for the model is that the company wants to keep its production costs below or equal to $75,000.