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Chapter 10: Problem 1

When you put a rock into a container of water, it raises the water level \(3 \mathrm{cm}\) If the container is a rectangular prism whose base measures \(15 \mathrm{cm}\) by \(15 \mathrm{cm},\) what is the volume of the rock?

Short Answer

Expert verified
The volume of the rock is 675 cm鲁.

Step by step solution

01

- Understand the problem

A rock is put into a rectangular prism container, raising the water level by 3 cm. You need to find the volume of the rock. The base of the container measures 15 cm by 15 cm.
02

- Recall the formula for volume

The volume of a rectangular prism is given by the formula: \[ V = \text{length} \times \text{width} \times \text{height} \]In this case, the height is the rise in water level, which is 3 cm, and the length and width are both 15 cm.
03

- Plug in the values

Substitute the known values into the volume formula: \[ V = 15 \text{ cm} \times 15 \text{ cm} \times 3 \text{ cm} \]
04

- Calculate the volume

Calculate the product of 15, 15, and 3: \[ V = 15 \times 15 \times 3 = 675 \text{ cm}^3 \]
05

- Conclusion

Thus, the volume of the rock is 675 cm鲁.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rectangular prism
A rectangular prism is a three-dimensional solid object which has six faces that are all rectangles. In geometry, it's important to understand that a rectangular prism has:
  • length
  • width
  • height
All these dimensions are perpendicular to each other. The base of our rectangular prism is a rectangle, and in our problem, the base measures 15 cm by 15 cm. This means the length and the width are equal in this particular case.
volume formula
The volume of any rectangular prism can be found using the formula:
\( V = \text{length} \times \text{width} \times \text{height} \)
Here,
  • V represents the volume
  • The length is one of the sides of the base.
  • The width is the other side of the base.
  • The height is the vertical distance from the base to the top surface
When we apply this formula to our problem, we know
  • Length = 15 cm
  • Width = 15 cm
  • Height (the rise in water level) = 3 cm
problem solving
Solving a geometry problem involves clear and straightforward steps. Here's what we did in our example problem:
  • We started by understanding the problem statement, identifying the dimensions given, and what we need to find out.
  • We recalled the volume formula for a rectangular prism: \( V = \text{length} \times \text{width} \times \text{height} \)
  • We substituted the given values into the formula: \[ V = 15 \text{cm} \times 15 \text{cm} \times 3 \text{cm} \]
  • We performed the multiplication to find the volume: \[ V = 675 \text{cm}^3 \]
Thus, our solution showed that the volume of the rock is 675 cm鲁. Each step in solving geometrical problems like these should be logical and ordered, ensuring all given information is used appropriately.

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Most popular questions from this chapter

Use this information to solve: Water weighs about 63 pounds per cubic foot, and a cubic foot of water is about 7.5 gallons. A child's wading pool has a diameter of 7 feet and is 8 inches deep. How many gallons of water can the pool hold? Round your answer to the nearest 0.1 gallon.

Find the ratio of the area of the circle inscribed in a square to the area of the circumscribed circle.

Refer to the table. $$\begin{array}{|l|l|l|l|} \hline \text { Metal } & \text { Density } & \text { Metal } & \text { Density } \\\\\hline \text { Aluminum } & 2.81 \mathrm{g} / \mathrm{cm}^{3} & \text { Nickel } & 8.89 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Copper } & 8.97 \mathrm{g} / \mathrm{cm}^{3} & \text { Platinum } & 21.40 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Gold } & 19.30 \mathrm{g} / \mathrm{cm}^{3} & \text { Potassium } & 0.86 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Lead } & 11.30 \mathrm{g} / \mathrm{cm}^{3} & \text { Silver } & 10.50 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Lithium } & 0.54 \mathrm{g} / \mathrm{cm}^{3} & \text { Sodium } & 0.97 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline\end{array}$$ Chemist Dean Dalton is given a clump of metal and is told that it is sodium. He finds that the metal has mass \(145.5 \mathrm{g}\). He places it into a nonreactive liquid in a square prism whose base measures \(10 \mathrm{cm}\) on each edge. If the metal is indeed sodium, how high should the liquid level rise?

Refer to the table. $$\begin{array}{|l|l|l|l|} \hline \text { Metal } & \text { Density } & \text { Metal } & \text { Density } \\\\\hline \text { Aluminum } & 2.81 \mathrm{g} / \mathrm{cm}^{3} & \text { Nickel } & 8.89 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Copper } & 8.97 \mathrm{g} / \mathrm{cm}^{3} & \text { Platinum } & 21.40 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Gold } & 19.30 \mathrm{g} / \mathrm{cm}^{3} & \text { Potassium } & 0.86 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Lead } & 11.30 \mathrm{g} / \mathrm{cm}^{3} & \text { Silver } & 10.50 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Lithium } & 0.54 \mathrm{g} / \mathrm{cm}^{3} & \text { Sodium } & 0.97 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline\end{array}$$ When ice floats in water, one-eighth of its volume floats above the water level and seven-eighths floats beneath the water level. A block of ice placed into an ice chest causes the water in the chest to rise \(4 \mathrm{cm} .\) The right rectangular chest measures \(35 \mathrm{cm}\) by \(50 \mathrm{cm}\) by \(30 \mathrm{cm}\) high. What is the volume of the block of ice?

About \(70 \%\) of Earth's surface is covered by water. If the diameter of Earth is about \(12,750 \mathrm{km},\) find the area not covered by water to the nearest \(100,000 \mathrm{km}^{2}\)
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