Calculating the Volume of the Rock

Oct 13, 2025 by talynt.org 138 views

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Introduction

In this problem, we are given a cubic tank with a side length of 2 meters. The tank contains a regular quadrilateral pyramid of rock with a base of 9 decimeters and a height of 10 decimeters. The water level in the tank is initially at 1.3 meters. Our task is to determine the water level in the tank after the rock is removed.

Calculating the Volume of the Rock

To solve this problem, we need to calculate the volume of the rock first. The rock is in the shape of a regular quadrilateral pyramid, which has a base area of 9 dm^2 and a height of 10 dm. The formula to calculate the volume of a pyramid is:

Volume = (1/3) × Base Area × Height

Substituting the given values, we get:

Volume = (1/3) × 9 dm^2 × 10 dm = 30 dm^3

Calculating the Volume of Water Displaced by the Rock

When the rock is removed from the tank, it displaces a volume of water equal to its own volume. Therefore, the volume of water displaced by the rock is also 30 dm^3.

Calculating the Initial Volume of Water in the Tank

The tank has a side length of 2 meters, which is equivalent to 200 dm. The water level in the tank is initially at 1.3 meters, which is equivalent to 130 dm. Therefore, the initial volume of water in the tank is:

Initial Volume = 200 dm × 130 dm = 26,000 dm^3

Calculating the Volume of Water Remaining in the Tank

After the rock is removed, the volume of water remaining in the tank is equal to the initial volume minus the volume of water displaced by the rock. Therefore, the volume of water remaining in the tank is:

Remaining Volume = 26,000 dm^3 - 30 dm^3 = 25,970 dm^3

Calculating the New Water Level in the Tank

The new water level in the tank can be calculated by dividing the remaining volume of water by the cross-sectional area of the tank. The cross-sectional area of the tank is:

Cross-Sectional Area = 200 dm × 200 dm = 40,000 dm^2

Therefore, the new water level in the tank is:

New Water Level = Remaining Volume / Cross-Sectional Area = 25,970 dm^3 / 40,000 dm^2 = 0.64925 m

Conclusion

In conclusion, after the rock is removed from the tank, the water level will rise to 0.64925 meters.

Final Answer

The final answer is 0.64925 meters.

Discussion

This problem requires a good understanding of geometry and volume calculations. The key concept here is to calculate the volume of the rock and the volume of water displaced by the rock, and then use these values to determine the new water level in the tank. The problem also requires the use of formulas and calculations to arrive at the final answer.

References

[1] "Geometry and Measurement" by [Author]
[2] "Mathematics for Engineers" by [Author]

Note: The references provided are fictional and for demonstration purposes only.

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Introduction

In our previous article, we discussed how to calculate the water level in a cubic tank after a rock is removed. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q: What is the formula to calculate the volume of a pyramid?

A: The formula to calculate the volume of a pyramid is:

Volume = (1/3) × Base Area × Height

Q: What is the base area of the rock in the problem?

A: The base area of the rock is 9 dm^2.

Q: What is the height of the rock in the problem?

A: The height of the rock is 10 dm.

Q: What is the volume of the rock in the problem?

A: The volume of the rock is 30 dm^3.

Q: What is the initial volume of water in the tank?

A: The initial volume of water in the tank is 26,000 dm^3.

Q: What is the volume of water displaced by the rock?

A: The volume of water displaced by the rock is 30 dm^3.

Q: What is the remaining volume of water in the tank after the rock is removed?

A: The remaining volume of water in the tank is 25,970 dm^3.

Q: How do we calculate the new water level in the tank?

A: We calculate the new water level in the tank by dividing the remaining volume of water by the cross-sectional area of the tank.

Q: What is the cross-sectional area of the tank?

A: The cross-sectional area of the tank is 40,000 dm^2.

Q: What is the new water level in the tank?

A: The new water level in the tank is 0.64925 m.

Q: What is the final answer to the problem?

A: The final answer to the problem is 0.64925 m.

Additional Tips and Tricks

When calculating the volume of a pyramid, make sure to use the correct formula and units.
When calculating the volume of water displaced by an object, make sure to use the correct formula and units.
When calculating the new water level in a tank, make sure to use the correct formula and units.
When working with different units, make sure to convert them correctly.

Common Mistakes to Avoid

Not using the correct formula to calculate the volume of a pyramid.
Not using the correct formula to calculate the volume of water displaced by an object.
Not using the correct formula to calculate the new water level in a tank.
Not converting units correctly.

Conclusion

In conclusion, calculating the water level in a cubic tank after a rock is removed requires a good understanding of geometry and volume calculations. By following the correct formulas and units, we can arrive at the correct answer. We hope this Q&A article has helped clarify any doubts and provided additional information on the topic.

Final Answer

The final answer is 0.64925 m.

References

[1] "Geometry and Measurement" by [Author]
[2] "Mathematics for Engineers" by [Author]

Note: The references provided are fictional and for demonstration purposes only.