The temperature of a chemical reaction oscillates between a low of 40 degrees C and a high of 100 degrees C. The temperature is at its lowest point when t = 0 and completes one cycle over a five-hour period.
(a) Sketch below the temperature, T, against the elapsed time, t, over a ten-hour period.
(b) Find the period, the amplitude, and the midline of the graph you drew in part (a).
Period is hours.
Amplitude is ___ degrees C.
Midline is T = ___ degrees C.
(a) I would love to sketch the temperature against elapsed time for you, but I'm just a bot and I don't have any artistic skills. However, I can describe it to you!
Imagine a roller coaster for temperatures. At t = 0, the temperature starts at the low point of 40 degrees C. Then, it gradually rises, reaches its peak at 100 degrees C, and then starts dropping again. It completes this cycle every five hours. Over a ten-hour period, you would see the temperature rise, reach its peak, and then drop back to the low point, before repeating the same pattern again.
(b) The period of the graph is given as 5 hours, since it completes one cycle every five hours.
The amplitude is the distance between the midline and either the highest or lowest point of the graph. From the information given, we know that the lowest point is 40 degrees C and the highest point is 100 degrees C. So, the amplitude is half the difference between these two values, which is (100 - 40) / 2 = 60/2 = 30 degrees C.
The midline is the line that represents the average or center value of the graph. Since the lowest point is 40 degrees C and the highest point is 100 degrees C, the midline would be halfway between these two values. So, the midline is (40 + 100) / 2 = 140 / 2 = 70 degrees C.
Therefore:
- Period is 5 hours.
- Amplitude is 30 degrees C.
- Midline is T = 70 degrees C.
(a) To sketch the graph, we need to divide the ten-hour period into two cycles of the temperature oscillations. Each cycle completes over a five-hour period, so we can divide the graph into two parts, each representing one cycle.
We can start by plotting the lowest temperature of 40 degrees Celsius at t = 0, and the highest temperature of 100 degrees Celsius at t = 2.5 hours (the middle point of the first five-hour cycle). Then, we mirror this pattern for the second five-hour cycle.
The sketch will look like a sinusoidal wave, with the temperature rising and falling between the two extremes. Since we are only given the lowest and highest points, we will assume that the rest of the graph completes smoothly between these points.
Here is a rough sketch of the graph:
```
^
100 - |
| /\
| / \
| / \
| / \
40 - |
__|______|______|______|______|______|__> t (time)
0 2.5 5 7.5 10
```
(b) The period of the graph is the time it takes for one complete cycle. From the given information, we know that one cycle completes over a five-hour period. Hence, the period is 5 hours.
The amplitude of the graph is the distance from the midline (the average of the highest and lowest points) to the highest or lowest point. From the sketch, we can see that the distance between the midline and either extreme point is 30 degrees Celsius. Therefore, the amplitude is 30 degrees Celsius.
The midline of the graph is found by averaging the highest and lowest temperatures. Given that the lowest temperature is 40 degrees Celsius and the highest temperature is 100 degrees Celsius, the midline can be calculated as follows:
Midline = (Highest + Lowest) / 2
= (40 + 100) / 2
= 140 / 2
= 70 degrees Celsius
Therefore, the midline is T = 70 degrees Celsius.
To sketch the temperature graph over a ten-hour period, we need to consider the given information. The temperature oscillates between a low of 40 degrees Celsius and a high of 100 degrees Celsius. It completes one cycle over a five-hour period, and it starts at its lowest point when t = 0.
(a) Sketching the temperature, T, against elapsed time, t, over a ten-hour period:
To sketch the graph, we can divide the ten-hour period into two equal parts of five hours each. Since one cycle is completed in five hours, the temperature graph will repeat itself in the second half.
Here's how the sketch would look:
| X
T(°C) | X
| X
100 X
| X
| X
| X
| X
| X
| X
-------------------------
t=0 5 hours 10
When t = 0, the temperature is at its lowest point, 40 degrees Celsius. As time progresses, the temperature gradually increases until it reaches 100 degrees Celsius. It stays at this maximum for the next five hours. Then, the temperature starts decreasing from 100 degrees Celsius back to 40 degrees Celsius. This completes one cycle.
(b) Finding the period, amplitude, and midline of the graph:
The period of the graph is the time it takes for one complete cycle. In this case, the period is five hours.
The amplitude represents half the distance between the highest and lowest points of the graph. In this case, we can calculate it as half the difference between the high point (100 degrees Celsius) and the low point (40 degrees Celsius):
Amplitude = (100 - 40) / 2 = 60 / 2 = 30 degrees Celsius.
The midline is the horizontal line that passes through the center of the graph, dividing it into two equal parts. In this case, the midline is the average of the high and low points:
Midline = (100 + 40) / 2 = 140 / 2 = 70 degrees Celsius.
Therefore, the period is 5 hours, the amplitude is 30 degrees Celsius, and the midline is T = 70 degrees Celsius.