A rectangular tank measure 4m by 2.5m by 10m. If the empty tank is to be filled in 10mins by a cylindrical pipe through which water is forced at a rate of 1 metre per second, find to 3 significant figures the diameter of the pipe in centimetres.
To find the diameter of the pipe in centimeters, we need to calculate the volume of the rectangular tank and the volume flow rate of the water from the pipe.
First, let's calculate the volume of the tank. It is given that the tank measures 4m by 2.5m by 10m, so its volume is:
Volume of the tank = Length × Width × Height
= 4m × 2.5m × 10m
= 100m³
Next, let's calculate the volume flow rate of the water. It is given that the tank is filled in 10 minutes by a cylindrical pipe through which water is forced at a rate of 1 meter per second. Since the pipe's rate of flow is given in meters per second, we need to convert minutes to seconds and multiply by the rate of flow:
Volume flow rate of the water = 1 meter per second × 60 seconds per minute
= 60 meters per minute
To convert the volume flow rate from meters per minute to cubic meters per minute, we need to multiply it by the cross-sectional area of the pipe. The formula for the cross-sectional area of a cylindrical pipe is:
Area = π × r²
Assuming the pipe has a constant diameter (d) along its length, the radius (r) will be half the diameter. Therefore, the cross-sectional area can be written as:
Area = π × (d/2)²
= π × (d²/4)
Since we want to find the diameter of the pipe in centimeters, we will convert the cross-sectional area to cubic centimeters per minute by multiplying it by 1 million (1 m³ = 1,000,000 cm³):
Volume flow rate of the water = (60 meters per minute) × (π × (d²/4)) × (1,000,000 cubic centimeters per cubic meter)
= 60 × π × (d²/4) × 1,000,000 cubic centimeters per minute
= 15,000,000 × π × (d²/4) cubic centimeters per minute
Finally, let's equate the volume of the tank to the volume flow rate of the water to find the diameter of the pipe:
Volume of the tank = Volume flow rate of the water
100m³ = 15,000,000 × π × (d²/4) cubic centimeters per minute
Simplifying the equation, we get:
100m³ = 3,750,000 × π × (d²/4) cubic centimeters per minute
Solving for d², we have:
d² = (100m³ / (3,750,000 × π)) × 4 cubic centimeters per minute
= 8 / π cubic centimeters per minute
Taking the square root of both sides to find d, we get:
d = √(8 / π) cubic centimeters per minute
Converting to centimeters and rounding to 3 significant figures, we find:
d = √(8 / π) × 100 cm/min ≈ 22.660 cm
Therefore, the diameter of the pipe is approximately 22.660 centimeters to 3 significant figures.