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?
−4.57 ≤ x ≤ 39.87
1.12 ≤ x ≤ 34.18
−4.57 ≤ x ≤ 1.12 and 34.18 ≤ x ≤ 39.87
0 ≤ x < 1.12 and 34.18 < x ≤ 39.87
The correct constraint is 1.12 ≤ x ≤ 34.18. This is because the graph of C(x) is below the horizontal line y = 75 (representing $75,000) between x = 1.12 and x = 34.18, indicating that production costs are below $75,000 for those values of x. The other options either include values of x where the production costs are above $75,000 or exclude values of x where the production costs are below $75,000.
To determine the constraint that is reasonable for the model, we need to identify the range of x values for which the production cost, C(x), is less than $75,000.
From the graph, we can see that the production cost is below $75,000 for x values between -4.57 and 1.12, and between 34.18 and 39.87. The production cost is above $75,000 for x values between 1.12 and 34.18.
Therefore, the constraint that is reasonable for the model is:
-4.57 ≤ x ≤ 1.12 and 34.18 ≤ x ≤ 39.87
So, the correct answer is option C: -4.57 ≤ x ≤ 1.12 and 34.18 ≤ x ≤ 39.87.