During a manufacturing process, a metal part in a machine is exposed to varying temperature conditions. The manufacturer of the machine recommends that the temperature of the machine part remain below 141°F. The temperature T in degrees Fahrenheit x minutes after the machine is put into operation is modeled by T = – 0.005x2 + 0.45x + 125. Will the temperature of the part ever reach or exceed 141°F? Use the discriminant of a quadratic equation to decide.
(1 point)
Responses
yes
yes
no
To determine if the temperature of the part will ever reach or exceed 141°F, we can set up the inequality T ≥ 141 and solve for x.
T = -0.005x^2 + 0.45x + 125 ≥ 141
Subtract 141 from both sides:
-0.005x^2 + 0.45x - 16 ≥ 0
To solve this inequality, we can use the quadratic formula. The discriminant of the quadratic equation is calculated as follows:
Discriminant = b^2 - 4ac
In this case, a = -0.005, b = 0.45, and c = -16.
Discriminant = (0.45)^2 - 4(-0.005)(-16)
= 0.2025 - 0.32
= -0.1175
Since the discriminant is negative, there will be no real solutions to this quadratic equation; therefore, the temperature of the part will never reach or exceed 141°F.
Therefore, the answer is no.