Water flows at the rate of 10 m/min from a cylindrical pipe 6 mm in diameter. How long will it take to fill up a conical vessel whose diameter at the base is 60 cm and depth 21 cm?

The correct option is 1 hour, 10 minutes.

To find the time taken to fill the conical vessel, we need to calculate both the volume of the conical vessel and the rate of flow of water from the cylindrical pipe, and then divide the volume of the vessel by the rate of flow.

First, let us write down the dimensions of both the pipe and the vessel, ensuring all units are consistent. We will convert all measurements into decimeters (dm) or centimeters (cm). Let's use centimeters (cm) for our calculations.

1. Dimensions and Volume of the Conical Vessel:
Diameter of the base of the cone = 60 cm
Radius of the base of the cone (R) = 60 / 2 = 30 cm
Depth (height) of the cone (H) = 21 cm

The formula for the volume of a cone is:
$$V_{\text{cone}}=\frac{1}{3}\pi R^{2}H$$
Substituting the values:
$$V_{\text{cone}}=\frac{1}{3}\times \pi \times {30}^2\times 21$$
$$V_{\text{cone}}=\frac{1}{3}\times \pi \times 900\times 21$$
$$V_{\text{cone}}=300\times 21\times \pi =6300\pi {\text{cm}}^3$$

2. Dimensions and Rate of Flow from the Cylindrical Pipe:
Diameter of the cylindrical pipe = 6 mm = 0.6 cm
Radius of the pipe (r) = 0.6 / 2 = 0.3 cm
Rate of flow of water = 10 m/min = 1000 cm/min

The volume of water flowing out of the pipe per minute is equivalent to the volume of a cylinder with radius r and length equal to the distance traveled by water in one minute (length h = 1000 cm).
$$\text{Volume of water per minute}=\pi r^{2}h$$
Substituting the values:
$$\text{Volume of water per minute}=\pi \times {\left(0.3\right)}^2\times 1000$$
$$\text{Volume of water per minute}=\pi \times 0.09\times 1000=90\pi {\text{cm}}^3\text{/min}$$

3. Calculation of Time Required:
The time required to fill the conical vessel is:
$$\text{Time}=\frac{\text{Total Volume of Conical Vessel}}{\text{Volume of water flowing per minute}}$$
$$\text{Time}=\frac{6300\pi }{90\pi }$$
$$\text{Time}=\frac{6300}{90}=70\text{minutes}$$

Converting 70 minutes into hours and minutes:
70 minutes = 60 minutes + 10 minutes = 1 hour, 10 minutes.