Alright, to solve this problem, we need to determine if the quadratic equation representing the temperature reaches or exceeds [tex]\(132^\circ F\)[/tex]. The equation given is:
[tex]\[ T = -0.005x^2 + 0.45x + 125 \][/tex]
We want to check if [tex]\( T \geq 132 \)[/tex]. To do so, we set up the inequality:
[tex]\[ -0.005x^2 + 0.45x + 125 \geq 132 \][/tex]
First, rearrange this to form a standard quadratic equation:
[tex]\[ -0.005x^2 + 0.45x + 125 - 132 \geq 0 \][/tex]
[tex]\[ -0.005x^2 + 0.45x - 7 \geq 0 \][/tex]
At this point, the quadratic equation we need to solve is:
[tex]\[ -0.005x^2 + 0.45x - 7 = 0 \][/tex]
Next, we'll use the discriminant to determine if this equation has any real roots. The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
In our equation, [tex]\(a = -0.005\)[/tex], [tex]\(b = 0.45\)[/tex], and [tex]\(c = -7\)[/tex]. Plugging in these values, we get:
[tex]\[ \Delta = (0.45)^2 - 4(-0.005)(-7) \][/tex]
[tex]\[ \Delta = 0.2025 - (4 \cdot 0.005 \cdot 7) \][/tex]
[tex]\[ \Delta = 0.2025 - 0.14 \][/tex]
[tex]\[ \Delta = 0.0625 \][/tex]
The discriminant [tex]\(\Delta\)[/tex] is [tex]\(0.0625\)[/tex], which is greater than zero. A positive discriminant indicates that the quadratic equation has two distinct real roots.
Since the discriminant is positive, the equation [tex]\( -0.005x^2 + 0.45x - 7 = 0 \)[/tex] has real solutions. This implies that the quadratic function [tex]\( -0.005x^2 + 0.45x + 125 \)[/tex] will indeed reach the value of 132 at some points.
Therefore, the temperature of the metal part in the machine will reach or exceed [tex]\(132^\circ F\)[/tex].
So, the answer is:
b. yes
