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Current Question (ID: 10437)

Question:

A tank is filled with water up to a height $h$ and the pressure at the bottom of the tank is $3P$ where $P$ is the atmospheric pressure. The height of a point from the bottom of the tank where pressure is $\frac{11P}{5}$ is:

Options:

1. $\frac{2h}{5}$
2. $\frac{3h}{5}$
3. $\frac{4h}{5}$
4. None of these

Solution:

1. Bottom pressure equation: The total pressure at the bottom ($3P$) is the sum of atmospheric pressure ($P$) and the pressure due to the water column ($\rho gh$). $3P = P + \rho gh \implies 2P = \rho gh$ 2. Pressure at a new height: Let $x$ be the height from the bottom where the pressure is $\frac{11P}{5}$. The height of the water column above this point is $(h-x)$. $\frac{11P}{5} = P + \rho g(h-x)$ Subtracting $P$ from both sides, we get: $\frac{11P}{5} - P = \rho g(h-x)$ $\frac{11P-5P}{5} = \rho g(h-x)$ $\frac{6P}{5} = \rho g(h-x)$ 3. Solve for $x$: Divide the second equation by the first equation. $\frac{\frac{6P}{5}}{2P} = \frac{\rho g(h-x)}{\rho gh}$ $\frac{6P}{5 \times 2P} = \frac{h-x}{h}$ $\frac{3}{5} = \frac{h-x}{h}$ $3h = 5(h-x)$ $3h = 5h - 5x$ $5x = 5h - 3h$ $5x = 2h \implies x = \frac{2h}{5}$ The correct option is 1.

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Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}