Question

The width of a rectangular prism is \(w\) centimetres. The height is \(2 \mathrm{cm}\) less than the width. The length is \(4 \mathrm{cm}\) more than the width. If the magnitude of the volume of the prism is 8 times the measure of the length, what are the dimensions of the prism?

Short answer

Width = 4 cm, Height = 2 cm, Length = 8 cm.

Step-by-step solution

- Identify the Given Information

Given, width of the rectangular prism is denoted as \(w\) cm. The height is 2 cm less than the width which can be written as \(w - 2\) cm. The length is 4 cm more than the width, so it is \(w + 4\) cm.

- Write the Volume Formula

The volume \(V\) of a rectangular prism is given by multiplying its length, width, and height: \(V = \text{length} \times \text{width} \times \text{height}\). Therefore, \(V = (w + 4) \times w \times (w - 2)\).

- Set Up the Volume Condition

The problem states that the volume of the prism is 8 times the measure of the length. The length is \(w + 4\), so the volume is \(8 \times (w + 4)\). This gives the equation: \((w + 4) \times w \times (w - 2) = 8(w + 4)\).

- Simplify the Equation

Divide both sides of the equation by \(w + 4\) (assuming \(w eq -4\)): \(w \times (w - 2) = 8\). This simplifies to \(w^2 - 2w = 8\).

- Solve the Quadratic Equation

Rewrite the equation as \(w^2 - 2w - 8 = 0\). Factorize it to get \((w - 4)(w + 2) = 0\). Therefore, \(w = 4\) or \(w = -2\). Since width must be a positive value, \(w = 4\).

- Determine the Dimensions

Substitute \(w = 4\) back into the expressions for length and height. The height is \(4 - 2 = 2\) cm and the length is \(4 + 4 = 8\) cm. Thus, the dimensions of the prism are width = 4 cm, height = 2 cm, and length = 8 cm.

Key concepts

volume of a rectangular prism

A rectangular prism is a 3D solid shape with six faces, all of which are rectangles. To determine the volume of a rectangular prism, you multiply its length, width, and height: \(V = \text{length} \times \text{width} \times \text{height}\). In our exercise, the width is denoted by \(w\), the height is \(w - 2\), and the length is \(w + 4\). So the volume can be expressed as: \[ V = (w + 4) \times w \times (w - 2)\]. Volume is a measure of how much space the prism occupies. In this problem, the prism's volume also equals 8 times its length, which adds a layer of complexity.

quadratic equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). Here, we derived a quadratic equation from our volume condition. Simplifying the given equation \( (w + 4) \times w \times (w - 2) = 8(w + 4)\) gives us: \[ w \times (w - 2) = 8\] which simplifies further to \[ w^2 - 2w = 8\]. This equation rephrases into standard quadratic form: \[ w^2 - 2w - 8 = 0\]. Solving quadratic equations typically involves factoring, using the quadratic formula, or completing the square.

problem-solving steps

When solving problems involving equations, especially in geometry, it's crucial to follow clear steps:

• First, identify all given information and write down the relationships among variables.
• Second, set up the necessary equations using known formulas (e.g., volume of a prism).
• Third, simplify the equations to solve for the unknowns.
• Finally, verify the solution by substituting it back into the original relationships.

Clear steps ensure that you don't skip important details or make errors in your calculations.

factoring polynomials

Factoring polynomials is a method of breaking down more complicated polynomials into simpler factors that, when multiplied, give back the original polynomial. For example, our quadratic equation: \[ w^2 - 2w - 8 = 0\]. We can factor it as: \[ (w - 4)(w + 2) = 0\]. Here, the factors are \(w - 4\) and \(w + 2\). Solving the equation, we find \(w = 4\) or \(w = -2\). Since width can't be negative, \(w = 4\). Factoring is a helpful technique and often the easiest way to solve quadratic equations when they can be factored cleanly.