A uniform cylindrical container is half filled with water. The height of the
cylinder is twice its diameter. The cylinder is gradually tilted until the water
touches the brim. At this instant, the container is inclined at which angle:
1.30
2.45
3.60
4.75
all to the vertical
Looking from the side, the cylinder has an outline of a rectangle, of height 2x and width x.
Since the cylinder is half-filled, at the point water touches the brim, there is as much water in the cylinder as there is air. So the water level divides the cylinder into two equal parts.
The final position of the cylinder is such that the diagonal is horizontal.
The angle the diagonal makes with the vertical side is tan-1(1/2)=26.565 degrees.
So the angle the axis of the cylinder makes with the vertical is 90-26.565=73.435 degrees, unfortunately it does not correspond to any one of the choices above.
Anchal, could you check if there is a typo in the question?
Well, that's quite a dilemma. Let's dive into the humor pool and find the answer!
You see, when the cylinder is half-filled with water, it's like a "H2Oway" rest stop for the water molecules. But when it's tilted, it's a "spill-a-coaster" ride!
Now, to determine the angle at which the water touches the brim, we need to do some "water-angleometry" (totally a real word).
Since the height of the cylinder is twice its diameter, we can imagine the cylinder as a very fancy top hat. And when you tilt a top hat, you become a tilt-a-trotter, making the water go "whoa, that's quite an angle!"
So, drumroll please, the answer is 4. 75 degrees! Because, you know, water likes to take things to the extreme, especially when it comes to angles.
Keep laughing and learning, my friend!
To find the angle at which the container is inclined when the water touches the brim, we can use trigonometry.
Let's denote:
- h as the height of the cylinder
- d as the diameter of the cylinder
Given that the height of the cylinder is twice its diameter, we can say that h = 2d.
We are told that the cylinder is half filled with water, which means that the water level reaches a height of h/2.
When the cylinder is tilted until the water touches the brim, we can imagine a right-angled triangle being formed. The height of this triangle is h/2, the base is d, and the hypotenuse is the inclined height of the cylinder when the water touches the brim.
Using the Pythagorean theorem, we can write the equation:
(h/2)^2 + d^2 = hypotenuse^2
Substituting h = 2d, we have:
((2d)/2)^2 + d^2 = hypotenuse^2
d^2 + d^2 = hypotenuse^2
2d^2 = hypotenuse^2
hypotenuse = sqrt(2d^2) = sqrt(2)d
Now, let's find the angle at which the container is inclined. The tangent of this angle can be found using the formula:
tan(angle) = (opposite/adjacent) = (h/2) / d = (2d/2) / d = 1
We need to find the angle whose tangent is 1. Taking the inverse tangent (arctan) of both sides:
angle = arctan(1)
Using a calculator, we find that the angle is approximately 45 degrees.
Therefore, the answer is 2. 45 degrees
To determine the angle at which the container is inclined when the water touches the brim, we can use trigonometry. Let's go step by step:
1. Start by visualizing the situation. We have a cylindrical container that is half filled with water. The height of the cylinder is twice its diameter. Let's assume the diameter of the cylinder is "d." So, the height of the cylinder would be 2d.
2. To find the angle of inclination, we need to consider the right triangle formed by the container's axis, the radius of the cylinder, and the inclined surface of the water.
3. The radius of the cylinder is half of its diameter. Therefore, the radius would be d/2.
4. Now, let's consider the inclined surface of the water. It will be a diagonal line connecting the base of the cylinder to the water's brim. Let's label this diagonal line as "x."
5. Since the height of the cylinder is twice its diameter, we can express the height as 2d. Considering the right triangle formed by the radius, height, and the inclined surface, we can see that the base of the triangle will be d/2 + d/2 = d. The height of the triangle will be 2d. And the hypotenuse (the inclined surface) will be x.
6. Now, we can use the trigonometric function "sine" to find the angle of inclination. The sine of an angle can be calculated as the ratio of the opposite side to the hypotenuse. In this case, the opposite side is the height (2d), and the hypotenuse is the inclined surface (x).
7. The formula for the sine function is sin(angle) = opposite/hypotenuse. Substituting the values, we have sin(angle) = (2d)/x.
8. Rearranging the equation, we can find "x" in terms of the angle: x = (2d)/(sin(angle)).
9. We know that at the instant the water touches the brim, the container is inclined at an angle where the height is equal to the radius (half of the cylinder's diameter). So, we can write the equation 2d = (d/2)/(sin(angle)). Simplifying this equation, we have 2 = 1/(2*sin(angle)).
10. Further simplifying, we get sin(angle) = 1/(4), which is equivalent to sin(angle) = 0.25.
11. To find the angle itself, we take the inverse sine of both sides: angle = arcsin(0.25).
12. Using a calculator or a trigonometric table, we can find the angle that has a sine of 0.25. The result is approximately 14.48 degrees.
So, the correct answer option from the given choices is not any of the provided angles. Rather, the inclined angle at the instant the water touches the brim is approximately 14.48 degrees.
It is given that height of the cylinder is twice that of its diameter. Using this we can imagine that when we tilt the cylinder until the water touches the brim, the height falls to half of its original height. Using formula
Sin x = p/h
Where, p = h/2 = D
and h= 2D
We get, sin x = 1/2
x = 30 degree