Question

Reducing the Size of a Candy Bar A jumbo chocolate bar with a rectangular shape measures 12 centimeters in length, 7 centimeters in width, and 3 centimeters in thickness. Due to escalating costs of cocoa, management decides to reduce the volume of the bar by \(10 \%\). To accomplish this reduction, management decides that the new bar should have the same 3 -centimeter thickness, but the length and width of each should be reduced an equal number of centimeters. What should be the dimensions of the new candy bar?

Short answer

The new dimensions of the candy bar are 11.55 cm in length, 6.55 cm in width, and 3 cm in thickness.

Step-by-step solution

Calculate the Original Volume

First, find the volume of the original chocolate bar using the given dimensions: length = 12 cm, width = 7 cm, and thickness = 3 cm. Use the volume formula for a rectangular prism: \[ V = l \times w \times h \] where \( l, w, \) and \( h \) are the length, width, and thickness respectively. \[ V = 12 \times 7 \times 3 = 252 \text{ cubic centimeters} \]

Calculate the Reduced Volume

Since the volume is to be reduced by 10%, find the new volume by calculating 90% of the original volume. \[ V_{\text{new}} = 0.9 \times 252 = 226.8 \text{ cubic centimeters} \]

Set up the Equation for the New Dimensions

Let the reduction in both the length and the width be \( x \) centimeters. The new dimensions will be \( 12 - x \) for the length and \( 7 - x \) for the width, with the thickness remaining the same at 3 cm. The new volume equation is: \[ (12 - x) \times (7 - x) \times 3 = 226.8 \]

Simplify the Equation

Divide both sides of the equation by 3 to simplify: \[ (12 - x) \times (7 - x) = 75.6 \]

Expand and Solve the Quadratic Equation

Expand the left side of the equation and rearrange to form a quadratic equation: \[ (12 - x)(7 - x) = 75.6 \] \[ 84 - 19x + x^2 = 75.6 \] \[ x^2 - 19x + 8.4 = 0 \]

Solve the Quadratic Equation

Solve the quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For \( a = 1, b = -19, \) and \( c = 8.4 \): \[ x = \frac{19 \pm \sqrt{361 - 33.6}}{2} = \frac{19 \pm \sqrt{327.4}}{2} \] \[ x = \frac{19 \pm 18.1}{2} \] \[ x_1 = 18.55 \text{ and } x_2 = 0.45 \]

Select the Valid Solution

Given that \( x \) representing the reduction must be less than the original dimensions \( x_1 = 18.55 \) is not valid, select \( x = 0.45 \).

Calculate the New Dimensions

Subtract \( x = 0.45 \) cm from the original length and width to find the new dimensions: \[ \text{New Length} = 12 - 0.45 = 11.55 \text{ cm} \] \[ \text{New Width} = 7 - 0.45 = 6.55 \text{ cm} \]

Key concepts

Volume Calculation

Volume calculation is a crucial skill, especially when it comes to understanding the size and capacity of objects in real life. It helps in many aspects like designing, manufacturing, and even in everyday activities like cooking. To find the volume of a rectangular prism, use the formula: \[ V = l \times w \times h \] where \( l \) is length, \( w \) is width, and \( h \) is height (or thickness). In our exercise, the original dimensions of the chocolate bar are 12 cm in length, 7 cm in width, and 3 cm in thickness. So, we calculate the volume as follows: \[ V = 12 \times 7 \times 3 = 252 \text{ cubic centimeters} \] This calculated value helps us understand how much space the chocolate bar occupies.

Rectangular Prism

A rectangular prism is a 3D shape with six faces, all of which are rectangles. It's a common shape in many everyday objects, such as boxes, bricks, and chocolate bars. When calculating the volume of a rectangular prism, it’s essential to remember that all faces are rectangular, and the opposite faces are equal in area. So, if you have the dimensions (length, width, and height), you can easily find the volume. For our chocolate bar, the dimensions are:

• Length: 12 cm
• Width: 7 cm
• Thickness: 3 cm

By reducing the chocolate bar's volume by 10%, we have to adjust its length and width proportionally while keeping the thickness constant.

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations of the form: \[ ax^2 + bx + c = 0 \] The solution for \( x \) is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula is derived from completing the square method and is essential when factoring is not straightforward. For our problem:

• We derived the equation \( x^2 - 19x + 8.4 = 0 \)
• Using the quadratic formula where \( a = 1 \), \( b = -19 \), and \( c = 8.4 \), we found two potential solutions, \( 18.55 \) and \( 0.45 \)

Since 18.55 is greater than our original dimensions, it’s not valid. Therefore, \( x = 0.45 \) is the reduction needed in length and width.

Dimensional Reduction

Dimensional reduction in this context refers to reducing the size of the chocolate bar’s length and width by the same amount to achieve a desired volume. Given that the thickness remains constant at 3 cm, we need to solve for \( x \), the amount to reduce both the length and width by. Here's how we calculated it:

• Start with the new volume: 226.8 cubic centimeters
• Set the equation for new dimensions with thickness fixed: \( (12 - x) \times (7 - x) \times 3 = 226.8 \)
• Simplify to form a quadratic equation: \( x^2 - 19x + 8.4 = 0 \)
• Apply quadratic formula to solve for \( x \)
• Verify the suitable reduction value, which is \( x = 0.45 \)

Finally, reduce the original dimensions: \[ \text{New Length} = 12 - 0.45 = 11.55 \text{ cm} \] \[ \text{New Width} = 7 - 0.45 = 6.55 \text{ cm} \]