technical machinist is asked to build a cubical steel tank that will hold 555L of water.
Calculate in meters the smallest possible inside length of the tank. Round your answer to the nearest 0.01m.
The Correct Answer and Explanation is :
Solution:
We are given a cubical steel tank that must hold 555 liters (L) of water. Since 1 liter = 0.001 cubic meters (m³), the volume of the tank in cubic meters is:
[
555 \times 0.001 = 0.555 \text{ m}^3
]
Since the tank is cubical, its volume is given by:
[
V = s^3
]
where ( s ) is the side length of the cube. Setting up the equation:
[
s^3 = 0.555
]
To find ( s ), take the cube root of 0.555:
[
s = \sqrt[3]{0.555}
]
Using a calculator:
[
s \approx 0.8278 \text{ m}
]
Rounding to the nearest 0.01 m:
[
s \approx 0.83 \text{ m}
]
Explanation:
The problem involves designing a cubical tank with the smallest possible inside length that can hold 555 liters of water. Since the tank is cubical, all sides are equal, and the volume formula simplifies to:
[
V = s^3
]
where ( s ) is the length of one side. The first step is converting liters to cubic meters because SI units are required for accurate calculations. Since 1 liter = 0.001 m³, multiplying by 555 gives 0.555 m³ as the tank’s volume.
To determine the side length, we take the cube root of 0.555:
[
s = \sqrt[3]{0.555} \approx 0.8278
]
which rounds to 0.83 m to the nearest hundredth. This means the smallest possible inside length of the tank is 0.83 meters.
This calculation is important in machining and fabrication to ensure material efficiency and accuracy. By knowing the correct dimensions, a machinist can select appropriate steel sheets, minimize waste, and ensure the tank meets required specifications. The accuracy of cube root calculations ensures the tank holds exactly 555L without excess material usage or unnecessary adjustments during manufacturing.