Question

pretest: quadratic relationships
which of the following statements is true for this situation?
a. the materials temperature reaches a maximum at 25 hours.
b. the materials temperature reaches a minimum at 15 hours.
c. the material returned to its initial temperature after 40 hours.
d. the materials initial temperature is -30 degrees celsius.

Explanation:

Step1: Analyze the parabola's direction and vertex

The graph is a parabola opening upwards (since the coefficient of \(x^2\) is positive), so it has a minimum point (vertex), not a maximum. So options A and B (mentioning maximum or wrong minimum time) can be eliminated.

Step2: Check initial temperature (x=0)

At \(x = 0\) (initial time), the \(y\)-value (temperature) is 0 (from the graph, it crosses the y-axis at (0,0)), so option D (initial temp -30) is wrong.

Step3: Check temperature at x=40 and x=0

At \(x = 0\), \(y = 0\). At \(x = 40\), we can see from the graph (or symmetry, since the vertex is around x=15 - 25? Wait, the roots: the parabola crosses x-axis at 0 and 30? Wait, no, looking at the graph, when x=0, y=0; and it touches or crosses? Wait, the graph has a vertex (minimum) and then goes up. Wait, actually, the parabola has roots at x=0 and x=30? Wait, no, when x=30, y=0? Wait, the graph shows that at x=30, y=0. Wait, but the question is about x=40? Wait, no, let's re-examine. Wait, the initial temperature is at x=0, y=0. Now, check option C: "The material returned to its initial temperature after 40 hours." Wait, maybe I misread. Wait, the parabola: when x=0, y=0; then it goes down to a minimum, then up. Wait, at x=30, y=0? Wait, the graph shows that at x=30, y=0. Wait, but the option C says 40 hours. Wait, maybe I made a mistake. Wait, no, let's check each option again:

- Option A: Parabola opens up, so it has a minimum, not maximum. So A is wrong.
- Option B: The minimum (vertex) is around x=15? Wait, the vertex is between x=10 and x=20, maybe x=15? Wait, no, the graph's vertex: looking at the grid, each square is, say, 5 units? Wait, the y-axis: from -200 to 200, with 50-unit marks. The x-axis: 0,10,20,30,40. The vertex is at y=-200, around x=15? Wait, but the problem is, the parabola opens up, so minimum at vertex. But option B says minimum at 15 hours. Wait, but let's check the other options. Option D: Initial temperature (x=0) is y=0, not -30. So D is wrong. Option C: Initial temperature is 0 (x=0, y=0). Now, when x=40, what's y? The graph at x=40 is high, but wait, maybe the parabola is symmetric? Wait, no, the roots are at x=0 and x=30? Wait, no, at x=30, y=0? Wait, the graph shows that at x=30, y=0. Wait, but option C says 40 hours. Wait, maybe I messed up. Wait, no, let's re-express:

Wait, the key is:

- A: Maximum? No, opens up, so minimum. So A wrong.
- B: Minimum at 15? Let's see the vertex. The vertex is at x=15? The graph's minimum is around x=15 (since between x=10 and x=20, the lowest point). But wait, the parabola's vertex is the minimum. But let's check the y-value at x=15: it's the minimum. But option B says "reaches a minimum at 15 hours" – is that correct? Wait, but earlier I thought maybe roots at 0 and 30, so vertex at x=15 (midpoint of 0 and 30). Yes! Because for a parabola \(y = ax(x - 30)\) (since roots at 0 and 30), the vertex is at \(x = \frac{0 + 30}{2} = 15\). So the minimum is at x=15. Wait, but earlier I thought option B was wrong, but maybe I was wrong. Wait, no, let's check the options again:

Option A: Maximum at 25? No, minimum at 15. So A wrong.

Option B: Minimum at 15 hours. Yes, because the vertex (minimum) is at x=15 (midpoint of 0 and 30). So B is correct? Wait, but wait, the graph: when x=15, it's the lowest point (minimum). So B is correct? Wait, but earlier I thought option C: "returned to initial temperature after 40 hours". Initial temperature is 0 (x=0, y=0). At x=40, what's y? The graph at x=40 is way above 0, so C is wrong. Option D: Initial temperature is 0, not -30. So D is wrong. Wait, but maybe I made a mistake in the roots. Wait, the graph crosses the x-axis at x=0 and x=30? Because at x=30, y=0. So the parabola is \(y = ax(x - 30)\). Then the vertex is at x=15, which is the midpoint. So the minimum is at x=15, so option B is correct. Wait, but the original options:

Wait, the options are:

A. Maximum at 25 – wrong (minimum)

B. Minimum at 15 – correct (vertex at x=15)

C. Returned to initial (0) at 40 – at x=40, y is high, so wrong

D. Initial temp -30 – at x=0, y=0, so wrong

So the correct answer is B? Wait, but wait, the graph: when x=0, y=0; x=30, y=0. So vertex at x=15, y=-200 (minimum). So option B says "The material’s temperature reaches a minimum at 15 hours" – that's correct.

Wait, but earlier I thought maybe C, but no. So the correct answer is B.

Answer:

B. The material’s temperature reaches a minimum at 15 hours.