The Length Of A Box Is 5 Meters Less Than Twice The Width, The Height Is 4 Meters More Than Three Times The Width, And The Volume Is 520 Cubic Meters. What Equation Can Be Used To Find The Height?

by ADMIN 197 views

In this article, we will delve into a fascinating problem involving the dimensions of a box and its volume. The problem presents us with a scenario where the length, width, and height of a box are related in specific ways, and we are tasked with identifying the equation that can be used to determine the height of the box. This problem falls under the realm of mathematics, specifically algebra and geometry, and requires us to translate word problems into mathematical expressions and equations. Understanding these concepts is crucial for various applications in engineering, physics, and everyday problem-solving.

The problem states the following:

  • The length of a box is five meters less than twice the width.
  • The height is four meters more than three times the width.
  • The box has a volume of 520 cubic meters.

The question asks which of the following equations can be used to find the height of the box.

To tackle this problem effectively, let's break it down into smaller, manageable parts. The key is to represent the unknown dimensions of the box using variables and then translate the given relationships into algebraic expressions. We can then use the formula for the volume of a box to form an equation. Let's define our variables:

  • Let w represent the width of the box (in meters).
  • Let l represent the length of the box (in meters).
  • Let h represent the height of the box (in meters).

Now, let's translate the given information into algebraic expressions:

  1. "The length of a box is five meters less than twice the width" translates to:
    • l = 2w - 5
  2. "The height is four meters more than three times the width" translates to:
    • h = 3w + 4
  3. "The box has a volume of 520 cubic meters." The volume of a box is given by the formula:
    • V = l * w * h
    • Since the volume is 520 cubic meters, we have:
      • 520 = l * w * h

Now, we have three equations:

  1. l = 2w - 5
  2. h = 3w + 4
  3. 520 = l * w * h

Our goal is to find an equation that can be used to find the height (h) of the box. To do this, we can substitute the expressions for l and h from equations 1 and 2 into equation 3. This will give us an equation in terms of w only.

Substituting l = 2w - 5 and h = 3w + 4 into 520 = l * w * h, we get:

520 = (2w - 5) * w * (3w + 4)

This equation relates the width w to the volume of the box. However, the question asks for an equation that can be used to find the height of the box. To do this, we need to manipulate the equation to isolate h. Since h = 3w + 4, we can express w in terms of h:

h = 3w + 4

Subtract 4 from both sides:

h - 4 = 3w

Divide both sides by 3:

w = (h - 4) / 3

Now we have an expression for w in terms of h. We can substitute this expression back into the equation 520 = (2w - 5) * w * (h). First, let's substitute w = (h - 4) / 3 into l = 2w - 5:

l = 2 * ((h - 4) / 3) - 5

Simplify the expression for l:

l = (2h - 8) / 3 - 5

l = (2h - 8 - 15) / 3

l = (2h - 23) / 3

Now we have expressions for both l and w in terms of h. We can substitute these into the volume equation:

520 = l * w * h

520 = ((2h - 23) / 3) * ((h - 4) / 3) * h

Multiply both sides by 9 (3 * 3) to eliminate the fractions:

4680 = (2h - 23) * (h - 4) * h

This equation directly relates the height h to the volume of the box. We can use this equation to solve for h. This is the equation we were looking for.

Based on our calculations, the equation that can be used to find the height of the box is:

4680 = (2h - 23) * (h - 4) * h

This equation is a cubic equation in h, which means it might have up to three real solutions. However, in the context of this problem, only the positive solution makes sense, as the height of the box cannot be negative.

This problem highlights the importance of being able to translate word problems into mathematical expressions and equations. This skill is fundamental in various fields, including science, engineering, and finance. By carefully reading the problem statement, identifying the unknowns, and representing the relationships between them using algebraic expressions, we can effectively solve complex problems.

Here are some additional tips for solving similar problems:

  • Read the problem carefully: Understand what the problem is asking before you start trying to solve it.
  • Identify the unknowns: Determine what quantities you need to find.
  • Assign variables: Represent the unknowns with variables.
  • Translate the word problem into algebraic expressions: Use the given information to write equations that relate the variables.
  • Solve the equations: Use algebraic techniques to solve for the unknowns.
  • Check your answer: Make sure your answer makes sense in the context of the problem.

In conclusion, we have successfully identified the equation that can be used to find the height of the box. The process involved breaking down the problem, representing the dimensions using variables, translating the given relationships into algebraic expressions, and using the formula for the volume of a box to form an equation. By substituting and simplifying, we arrived at the equation 4680 = (2h - 23) * (h - 4) * h, which can be used to solve for the height h. This problem demonstrates the power of algebra in solving real-world problems and the importance of carefully translating word problems into mathematical expressions.

FAQ About Box Dimensions and Volume Relationships

What is the formula for the volume of a box?

The formula for the volume of a box (also known as a rectangular prism) is given by:

V = l * w * h

Where:

  • V is the volume
  • l is the length
  • w is the width
  • h is the height

This formula states that the volume of a box is equal to the product of its length, width, and height.

How do you translate word problems into algebraic expressions?

Translating word problems into algebraic expressions involves carefully reading the problem statement and identifying the key information. Here are some steps you can follow:

  1. Read the problem carefully: Understand what the problem is asking before you start trying to solve it.
  2. Identify the unknowns: Determine what quantities you need to find.
  3. Assign variables: Represent the unknowns with variables (e.g., x, y, z).
  4. Look for key words and phrases: Certain words and phrases indicate mathematical operations. For example:
    • "Less than" or "decreased by" indicates subtraction.
    • "More than" or "increased by" indicates addition.
    • "Times" or "product" indicates multiplication.
    • "Quotient" indicates division.
  5. Write the expressions: Use the variables and the mathematical operations to write algebraic expressions that represent the relationships described in the problem.

Why is it important to understand word problems in mathematics?

Understanding word problems is crucial in mathematics because they allow us to apply mathematical concepts to real-world situations. Word problems help us develop critical thinking and problem-solving skills, which are essential in various fields, including science, engineering, finance, and everyday life. By learning how to translate word problems into mathematical expressions and equations, we can effectively analyze and solve complex problems.

In summary, understanding the relationship between box dimensions and volume involves translating word problems into mathematical equations. This requires careful reading, identifying unknowns, assigning variables, and using algebraic expressions. The ability to solve these types of problems is crucial for various applications and develops critical thinking skills. By following the steps outlined in this article, you can effectively tackle similar problems and enhance your mathematical abilities.