Exams › JEE Advanced › Physics

Water leaks from the bottom of an inverted cone of semi-vertical angle 45 degrees at a rate of 2 m³/s. When the height of water in the cone is 2 m, find the rate d (in m/s) at which the circumference of the water surface is changing. What is |5d|?

1. 4
2. 5
3. 6
4. 7

Correct answer: 5

Solution

Since semi-vertical angle = 45 deg, r = h always. Volume V = (1/3) pi h³. Differentiating: dV/dt = pi h² * dh/dt. Water leaks at -2 m³/s: -2 = pi * 4 * dh/dt → dh/dt = -1/(2*pi). Circumference C = 2*pi*h, so dC/dt = 2*pi * dh/dt = 2*pi * (-1/(2*pi)) = -1 m/s. So d = -1, and |5d| = 5.

Related JEE Advanced Physics questions

• Rain is falling at an angle α with respect to the vertical at a velocity of 10 m/s. A girl runs in the opposite direction to the rain at a speed of 8 m/s and observes that the rain appears to make an angle β with the vertical. What is the relationship between α and β?
• A vector A is turned by a very small angle Δθ radians (where Δθ is much less than 1) to form a new vector B. What is the magnitude of the difference |B − A|?
• A swimmer moves through water at a speed of v relative to the water. To cross a river of width d, which flows at a speed u (where u < v), the downstream displacement caused by the river's flow is x. Which of the following is accurate?
• A particle is launched from a point with an initial speed u at an angle θ above the horizontal. At a specific point during its motion, its velocity becomes perpendicular to its original direction. What is the time taken to reach this point?
• Two objects, each with a mass of M, are positioned at a fixed distance of 2L apart. A smaller particle of mass m is launched from the midpoint of the line connecting their centers, moving in a direction perpendicular to this line. Given the gravitational constant G, which of the following statements is accurate?
• A central force is given as F(r) = −k / rⁿ, where k is a constant. What condition must n satisfy for a circular orbit to remain stable?

⚔️ Practice JEE Advanced Physics free + battle 1v1 →