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?
To determine whether the temperature of the part will reach or exceed 141°F, we need to solve the equation T = 141 and see if there are any solutions.
The given equation is T = -0.005x^2 + 0.45x + 125.
Setting T = 141, we have:
141 = -0.005x^2 + 0.45x + 125
Rearranging this equation, we get:
0 = -0.005x^2 + 0.45x + 125 - 141
0 = -0.005x^2 + 0.45x - 16
Now we have a quadratic equation in terms of x. To solve this equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = -0.005, b = 0.45, and c = -16. Plugging these values into the formula, we get:
x = (-0.45 ± √(0.45^2 - 4 * -0.005 * -16)) / (2 * -0.005)
Simplifying further, we have:
x = (-0.45 ± √(0.2025 - 0.128)) / (-0.01)
x = (-0.45 ± √0.0745) / (-0.01)
Calculating the values inside the square root, we have:
x = (-0.45 ± 0.2729) / (-0.01)
Now let's evaluate the two possible solutions:
x1 = (-0.45 + 0.2729) / (-0.01) = -17.29 / (-0.01) = 1729
x2 = (-0.45 - 0.2729) / (-0.01) = -7221 / (-0.01) = 722100
Since x represents time in minutes, we can conclude that the temperature will never reach or exceed 141°F because the positive solution, 1729 minutes, is much larger than any reasonable time frame for the manufacturing process.
Therefore, the temperature of the part will never reach or exceed 141°F.