To tackle the problem involving the faulty barometer, we need to analyze the relationship between the true readings and the faulty readings. Let's break this down step by step.
Determining the Total Length of the Barometer Tube
First, we need to establish a linear relationship between the true barometer readings and the faulty barometer readings. We have two pairs of readings:
- True reading: 73 cm, Faulty reading: 60 cm
- True reading: 75 cm, Faulty reading: 70 cm
To find the total length of the barometer tube, we can use the concept of slope (m) in a linear equation. The formula for slope between two points (x1, y1) and (x2, y2) is:
m = (y2 - y1) / (x2 - x1)
Here, let’s assign:
- x1 = 73, y1 = 60
- x2 = 75, y2 = 70
Now, substituting these values into the slope formula:
m = (70 - 60) / (75 - 73) = 10 / 2 = 5
This means for every 1 cm increase in the true reading, the faulty reading increases by 5 cm. Now, we can express the relationship between the true reading (T) and the faulty reading (F) as:
F = 5(T - 73) + 60
To find the total length of the barometer tube, we need to determine the maximum true reading, which is 75 cm. Plugging this into our equation:
F = 5(75 - 73) + 60 = 5(2) + 60 = 10 + 60 = 70 cm
Thus, the total length of the barometer tube is 75 cm, as it corresponds to the maximum true reading.
Finding the True Reading for a Faulty Reading of 69.5 cm
Next, we want to find the true reading when the faulty barometer reads 69.5 cm. We can rearrange our earlier equation:
69.5 = 5(T - 73) + 60
Now, let’s solve for T:
69.5 - 60 = 5(T - 73)
9.5 = 5(T - 73)
T - 73 = 9.5 / 5
T - 73 = 1.9
T = 74.9 cm
So, when the faulty barometer reads 69.5 cm, the true reading is approximately 74.9 cm.
Calculating the Faulty Reading for a True Reading of 74 cm
Finally, we need to determine the faulty barometer reading when the true reading is 74 cm. We can use our established equation again:
F = 5(74 - 73) + 60
Calculating this gives:
F = 5(1) + 60 = 5 + 60 = 65 cm
Therefore, when the true reading is 74 cm, the faulty barometer reads 65 cm.
Summary of Findings
- Total length of the barometer tube: 75 cm
- True reading for a faulty reading of 69.5 cm: 74.9 cm
- Faulty reading for a true reading of 74 cm: 65 cm
This structured approach allows us to clearly understand the relationships and calculations involved in working with the faulty barometer. If you have any further questions or need clarification on any of the steps, feel free to ask!
