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Chapter 8: Problem 51

A mercury barometer reads \(75 \mathrm{~cm}\). If the tube be inclined by \(60^{\circ}\) from vertical, the length of mercury in the tube will be (A) \(37.5 \mathrm{~cm}\) (B) \(150 \mathrm{~cm}\) (C) \(\frac{75 \sqrt{3}}{2} \mathrm{~cm}\) (D) \(100 \mathrm{~cm}\)

Short Answer

Expert verified
The length of mercury in the tube when inclined by 60 degrees from the vertical is 150 cm. The correct answer is (B) 150 cm.

Step by step solution

01

Visualize the problem

Draw the initial vertical position of the mercury barometer. The length of mercury is 75 cm. Then, draw the inclined position where the tube is inclined by 60 degrees from the vertical. Add a horizontal line from the top of the mercury in the inclined position to the vertical tube. This will create a right triangle.
02

Name the sides of the right triangle

Let's call the vertical side of the triangle "a", the horizontal side "b", and the inclined side (the length of mercury in the inclined position) "c". The angle given in the problem is 60 degrees, which will be between "a" and "c".
03

Use trigonometry to find the length of the inclined mercury

In a right triangle, we can use trigonometric functions. In this case, we can use the cosine function since we know the angle and the vertical side (75 cm). The cosine function is defined as: \[\cos(\theta) = \frac{adjacent}{hypotenuse}\] Since the adjacent side is "a" and the hypotenuse is "c", we can write: \[\cos(60^{\circ}) = \frac{a}{c}\] We know the vertical side "a" is 75 cm, so: \[\cos(60^{\circ}) = \frac{75}{c}\]
04

Solve for c

Now, we need to solve the equation to find the length of the inclined mercury "c": \[\frac{1}{2} = \frac{75}{c}\] \[c = \frac{75}{\frac{1}{2}}\] \[c = 150\]
05

Conclusion

The length of mercury in the tube when inclined by 60 degrees from the vertical is 150 cm. The correct answer is (B) 150 cm.

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