The graph of the function P(x) = −0.34x2 + 12x + 62 is shown. The function models the profits, P, in thousands of dollars for a tire company, 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 profits at or above $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 constraint should be: 1.12 ≤ x ≤ 34.18. This is because this is the range of x-values for which the profits, P, are greater than or equal to $75,000. We can see from the graph that the profits start to dip below $75,000 outside of this range. The other options either include x-values where the profits are too low or exclude x-values where the profits are above $75,000.
To determine which constraint is reasonable for the model, we need to identify the range of x values for which the profits, P, are at or above $75,000.
Looking at the graph, we can see that the profits are at or above $75,000 between the x-values of 1.12 and 34.18. This corresponds to the range 1.12 ≤ x ≤ 34.18.
Therefore, the reasonable constraint for the model is:
1.12 ≤ x ≤ 34.18.