Question

A rectangular box has sides 3,4 , and \(x\) and a volume of 18 . What is the value of \(x\) ?

Short answer

The value of x is \(\frac{3}{2}\).

Step-by-step solution

Write the equation for the volume of the rectangular box

The equation for the volume of the rectangular box is: Volume = length * width * height In our case: 18 = 3 * 4 * x

Simplify the equation

First, we can simplify the equation by multiplying 3 and 4: 18 = 12 * x

Solve for x

Now, we need to solve for x. To do this, we can divide both sides of the equation by 12: \(x = \frac{18}{12}\)

Simplify the fraction to get the value of x

To simplify the fraction, we can reduce it to the lowest terms: \(x = \frac{18}{12}= \frac{3\cdot6}{3\cdot4}\) \(x = \frac{3}{2}\) Therefore, the value of x is \(\frac{3}{2}\).

Key concepts

Volume Calculation

Volume is a key concept in geometry, especially when dealing with three-dimensional shapes like rectangular boxes. Volume refers to the amount of space an object occupies. To find the volume of a rectangular box, we use the formula: \[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \] In our exercise, you were given three dimensions: two known sides of the rectangular box (3 and 4) and one unknown dimension (\(x\)). The volume was given as 18 cubic units. By using the volume formula, you can set up an equation:- 18 = 3 \( \times \) 4 \( \times \) \(x\) This equation can be solved to find the missing dimension, \(x\). Understanding how to calculate volume is fundamental to solving this type of problem. It involves multiplying the length, width, and height, which is an essential skill in many real-world applications.

Rectangular Box

A rectangular box, also known as a cuboid, is a three-dimensional shape bounded by six rectangular faces. Each face meets another at a right angle, making it a very common shape in our everyday surroundings. When you think of a rectangular box, you need to remember:

• It has three dimensions: length, width, and height.
• The opposite faces are always equal in size.
• All angles are right angles (90 degrees).

In the context of solving geometry problems, knowing these properties helps you to apply the right formula for volume or surface area calculations. In this problem, you needed to apply these principles to find the missing measurement, \(x\), one of the dimensions of the box.

Solving Equations

Solving equations involves finding the value of an unknown variable that makes the equation true. In our case, it was finding the value of \(x\) in the equation 18 = 3 \( \times \) 4 \( \times \) \(x\). Here’s how you solve the equation for \(x\):- First, simplify both sides as much as possible.- Multiply the known elements (3 and 4) to reduce the equation to 18 = 12 \( \times \) \(x\).- Finally, isolate \(x\) by dividing both sides by 12. This gives you \(x = \frac{18}{12}\).Simple arithmetic operations and basic algebraic rules, such as maintaining equality by doing the same operation on both sides of the equation, are essential skills for solving such problems. These form the foundation for more advanced studies in mathematics.

Simplifying Fractions

Simplifying fractions is a crucial step in many mathematical calculations, helping you to express a fraction in its simplest form. In our solution, we dealt with \(x = \frac{18}{12}\) and simplified it to \(\frac{3}{2}\). Here’s how you can simplify a fraction:- Break down both the numerator (18) and the denominator (12) into their prime factors: 18 = 3 \(\times\) 6 and 12 = 3 \(\times\) 4.- Identify and cancel out any common factors. Here, the number 3 is common in both.- Divide both the numerator and the denominator by the greatest common divisor (GCD), which in this case is 6, to simplify further.The simplified fraction \(\frac{3}{2}\) helps in understanding the exact relationship between numbers with fewer, clearer terms. This process not only gives you the simplest form but also reveals important relationships and proportionality in real-life scenarios.