Question

What is the volume of a cone with a radius of $3\mathrm{cm}$ and a height of $5\mathrm{cm}$ (take $\pi$ to be $3.14$ and work to the nearest ${\mathrm{cm}}^3)$ ?

Short answer

47\text{ cm}^3

Step-by-step solution

Identify the formula for the volume of a cone

The volume of a cone can be calculated using the formula: $$V=\frac{1}{3}\pi r^{2}h$$ where $V$ is the volume, $r$ is the radius, and $h$ is the height.

Substitute the given values

Introduce the given values into the formula. Here, the radius $r=3\text{ cm}$ and the height $h=5\text{ cm}$. Therefore, we substitute these into the formula: $$V=\frac{1}{3}\pi (3^2)(5)$$

Calculate the area of the base

First, calculate the area of the base which is $\pi r^2$. Using $r=3\text{ cm}$: $$\pi (3^2)=\pi (9)$$ Using $\pi =3.14$: $$3.14\times 9=28.26{\text{ cm}}^2$$

Calculate the volume

Use the calculated area of the base to find the volume: $$V=\frac{1}{3}\times 28.26\times 5$$ Perform the multiplication: $$V=\frac{1}{3}\times 141.3$$ Finally, divide by 3: $$V=47.1{\text{ cm}}^3$$

Round the answer

Round the final answer to the nearest cubic centimeter: $$V\approx 47{\text{ cm}}^3$$

Key concepts

Volume of a Cone

In geometry, we often come across shapes like cones. A cone is a three-dimensional shape with a circular base that tapers smoothly to a point called the apex. The volume of a cone measures the space it occupies.

To find the volume, we use the formula: $$V=\frac{1}{3}\pi r^{2}h$$ Here, $r$ is the radius of the base, $h$ is the height, and $\pi$ is a constant approximately equal to 3.14. This formula shows that the volume of a cone depends on the area of the base and the height. We multiply the area of the base by the height, and then divide by three.

For example, if a cone has a radius of 3 cm and a height of 5 cm, we substitute these values into the formula: $$V=\frac{1}{3}\pi (3^2)(5)$$ Performing the calculations step-by-step, we find that: $$\pi (3^2)=\pi \times 9=3.14\times 9=28.26$$ Then, we continue to find the volume: $$V=\frac{1}{3}\times 28.26\times 5\approx 47{\text{ cm}}^3$$ By using this formula, we can easily determine the volume of any cone given its radius and height.

Cylinder Volume Formula

Understanding the volume of a cylinder is another crucial part of GMAT mathematics. A cylinder has two parallel circular bases connected by a curved surface. The volume measures the amount of space inside the cylinder.

The formula for the volume of a cylinder is: $$V=\pi r^{2}h$$ In this formula, $r$ indicates the radius of the bases and $h$ is the height. Note the similarity to the cone's volume formula—the main difference is the absence of the $\frac{1}{3}$ factor.

For instance, if a cylinder has a radius of 4 cm and a height of 7 cm, we substitute these values into the formula to find the volume: $$V=\pi (4^2)(7)$$ Step-by-step: $$\pi (4^2)=\pi \times 16=3.14\times 16=50.24{\text{ cm}}^2$$ Then, using this area, you multiply by the height: $$V=50.24\times 7=351.68{\text{ cm}}^3$$ This volume calculation is direct and useful for understanding how three-dimensional spaces are measured.

Mathematics for GMAT

Preparing for the GMAT involves mastering various mathematical concepts, including volume calculations of different shapes like cones and cylinders.

Here are some tips to help you excel in GMAT-related volume problems:

• Always carefully read the problem to identify which shape is involved.
• Write down the known values, such as the radius and height.
• Ensure you understand which formula to use—remember, cones and cylinders have slightly different formulas.
• Mental math is important; practice calculations to enhance your speed and accuracy.

These steps will not only help you tackle volume questions but also build your overall confidence in handling GMAT quantitative problems. Remember to keep practicing, and soon these calculations will become second nature.