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
no
To determine if the temperature of the part will ever reach or exceed 141°F, we can use the discriminant of the quadratic equation.
The quadratic equation representing the temperature T in degrees Fahrenheit is:
T = -0.005x^2 + 0.45x + 125
We can rewrite this equation in the form ax^2 + bx + c = 0:
-0.005x^2 + 0.45x + 125 - 141 = 0
Simplifying this equation, we get:
-0.005x^2 + 0.45x - 16 = 0
Now we can calculate the discriminant:
Discriminant = b^2 - 4ac = (0.45)^2 - 4(-0.005)(-16)
Discriminant = 0.2025 - 1.28
Discriminant = -1.0775
Since the discriminant is negative, it means that the quadratic equation has no real roots. In other words, the temperature of the part will never reach or exceed 141°F.
Therefore, the correct answer is: no.
To decide whether the temperature of the part will reach or exceed 141°F, we need to determine if the quadratic equation T = -0.005x^2 + 0.45x + 125 has any real solutions greater than or equal to 141°F.
The discriminant of a quadratic equation in the form Ax^2 + Bx + C = 0 is given by Δ = B^2 - 4AC.
In this case, the quadratic equation is T = -0.005x^2 + 0.45x + 125, so A = -0.005, B = 0.45, and C = 125.
The discriminant Δ is calculated as follows:
Δ = B^2 - 4AC
= (0.45)^2 - 4(-0.005)(125)
= 0.2025 + 2
Δ = 2.2025
Since the discriminant Δ is positive, there will be two distinct real solutions. Therefore, the temperature of the part will reach or exceed 141°F at some point during the manufacturing process.
So, the answer is: yes
To determine if the temperature of the part will ever reach or exceed 141°F, we can use the discriminant of the quadratic equation.
The discriminant, denoted as Δ, is calculated using the formula: Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.
In this case, the quadratic equation is T = -0.005x^2 + 0.45x + 125, with a = -0.005, b = 0.45, and c = 125.
Let's calculate the discriminant:
Δ = b^2 - 4ac
Δ = (0.45)^2 - 4(-0.005)(125)
Δ = 0.2025 + 25
Δ = 25.2025
Since the discriminant is positive (greater than zero), this means that the quadratic equation has two distinct real roots. In the context of the temperature equation, it implies that the temperature will reach the value of 141°F at some point.
Therefore, the answer to the question is yes, the temperature of the part will eventually reach or exceed 141°F.