Question

What is the volume of a sphere with a radius of \(3 \mathrm{~cm}\) (take \(\pi\) to be \(3.14\) and give your answer to the nearest full \(\left.\mathrm{cm}^3\right) ?\)

Short answer

113 cm³

Step-by-step solution

Recall the Formula for the Volume of a Sphere

The formula to find the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). Here, \( V \) represents the volume, \( r \) is the radius, and \( \pi \) is a constant approximately equal to 3.14.

Plug in the Given Radius

Substitute the given radius (3 cm) into the formula: \( V = \frac{4}{3} \pi (3)^3 \).

Calculate the Radius Cubed

Find \( 3^3 \), which is 27. So, the expression now is \( V = \frac{4}{3} \pi \times 27 \).

Multiply by \( \pi \) and Simplify

Multiply 27 by 3.14 (the given value of \( \pi \)): \( 27 \times 3.14 = 84.78 \).

Complete the Calculation

Now multiply by \( \frac{4}{3} \): \( \frac{4}{3} \times 84.78 \approx 113.04 \).

Round to the Nearest Whole Number

The result from Step 5 (113.04) is closest to 113 when rounded to the nearest whole number.

Key concepts

volume of a sphere

Understanding how to calculate the volume of a sphere is crucial for various math problems, especially in the GMAT quantitative section. The volume of a sphere can be determined using a specific formula that relates the radius of the sphere to its volume. This formula is:
\(V = \frac{4}{3} \pi r^3\).

In this formula, \(V\) stands for volume, \(r\) is the radius of the sphere, and \(\pi\) is a constant approximately equal to 3.14. Calculating the volume requires knowing the sphere's radius and correctly applying this formula.

mathematical formulas

Mathematical formulas are like the tools in a mathematician's toolkit. They help solve specific types of problems by providing a standardized method of calculation. For the volume of a sphere, the formula \(V = \frac{4}{3} \pi r^3\) is essential. Here's a breakdown:

• \(V\) is the volume of the sphere.
• \(r\) represents the radius.
• \(\pi\) is a constant, which you can approximate to 3.14 for practical purposes.

This formula combines constants and the radius to compute the sphere's volume accurately. Knowing how to correctly plug in values and perform arithmetic is vital for solving such problems.

problem-solving steps

Solving a math problem efficiently requires a systematic approach. Let's break down the steps to find the volume of a sphere:

1. **Recall the Formula:** Let's start with the formula for the volume of a sphere, \(V = \frac{4}{3} \pi r^3\).
2. **Plug in the Given Radius:** For this problem, substitute the radius with 3 cm: \(V = \frac{4}{3} \pi (3)^3\).
3. **Calculate the Radius Cubed:** Compute \(3^3\), which is 27. Now, the equation looks like this: \(V = \frac{4}{3} \pi \times 27\).
4. **Multiply by \(\pi\) and Simplify:** Multiply 27 by 3.14 (value of \(\pi\)): \(27 \times 3.14 = 84.78\).
5. **Complete the Calculation:** Finally, multiply by \frac{4}{3}: \(\frac{4}{3} \times 84.78 \approx 113.04\).
6. **Round to the Nearest Whole Number:** The result from step 5 is 113.04, which rounds to 113. Thus, the volume is approx. 113 \mathrm{cm}^3\.

GMAT quantitative section

The GMAT quantitative section tests your problem-solving skills, especially in math. It includes questions on arithmetic, algebra, and geometry. Understanding how to apply mathematical formulas, like the volume of a sphere, is essential for this section.

The volume of a sphere problem illustrates the type of geometry questions you might encounter. You'll need to:

• Recall and use specific formulas.
• Perform arithmetic operations accurately.
• Understand when and how to round your results.

Practicing with these types of problems will improve your confidence and ability to tackle the quantitative section effectively.