Question

Ken baked, frosted, and decorated a rectangular cake for the last Math Cllb meeting. The cake was 3 inches high, 12 inches wide, and 16 inches long. He centered the cake on a piece of cardboard whose rectangular top surface had been covered with aluminum foil, as shown in the figure below. (See Figure) At the Math Club meeting, Principal Gonzales cut the entire cake into pieces. Each piece is 2 inches wide, 2 inches long, and 3 inches high. What is the number of pieces Principal Gonzales cut the cake into? F. 16 G. 20 H. 28 J. 48 K. 96

Short answer

Answer: J. 48

Step-by-step solution

Determine the number of pieces in length and width

We need to determine how many pieces of 2 inches width will fit in a width of 12 inches, and how many pieces of 2 inches length will fit in a length of 16 inches. For the width: \(\frac{12}{2} = 6\) For the length: \(\frac{16}{2} = 8\) So the cake can be divided into 6 pieces in width and 8 pieces in length.

Calculate the total number of pieces

Now that we know the number of pieces in each direction, we can multiply them together to find the total number of pieces: Total number of pieces = 6 (width) × 8 (length) = 48

Final answer

The answer is J. 48. The cake was divided into 48 equal pieces.

Key concepts

Geometry in Everyday Life

Geometry, the branch of mathematics involving points, lines, planes, and figures, is an essential aspect of our daily experiences. As seen in the textbook exercise, the dimensions of the cake represent a real-world application of geometric principles. In particular, we examine the dimensions of three-dimensional shapes known as rectangular prisms, which in this case, is the shape of the cake.

A rectangular prism has six faces, all of which are rectangles. To describe such a shape, we often refer to its length, width, and height. Here, Ken's cake measures 3 inches high, 12 inches wide, and 16 inches long. These measurements are crucial as they are used in calculations to determine the volume of the shape, or in this scenario, to figure out how to correctly portion the cake.

Volume Calculation for Portion Control

Volume calculation is vital when dividing objects into smaller parts. In the context of Ken's cake, we are tasked to deduce how the cake can be equally portioned.

To calculate the volume of Ken's rectangular cake, we would use the formula for the volume of a rectangular prism: \( V = length \times width \times height \). However, when it comes to portioning, the focus changes from calculating the total volume to partitioning the existing volume into smaller volumes of equal size.

In this problem, Principal Gonzales cuts the cake into smaller rectangular prisms, each being 2 inches by 2 inches by 3 inches. Here, volume calculation aids us in determining the number of these smaller prisms (pieces of cake) that can fit into the larger one.

Fractions and Division in Problem Solving

Fractions and division are instrumental in the division of spaces or objects into equal parts. Looking at the cake-cutting exercise, we tackle the problem by dividing the cake's dimensions by the desired piece dimensions to ascertain the number of pieces.

For example, dividing the width of 12 inches by the 2-inch pieces gives us \( \frac{12}{2} = 6 \), indicating that 6 pieces can fit across the width. Similarly, by dividing the length of 16 inches by the 2-inch piece length, we get \( \frac{16}{2} = 8 \), meaning 8 pieces can be arranged along the length.

To find the total number of cake pieces, we multiply these numbers together. This use of division turns a complex problem into an accessible arithmetic task, demonstrating the practical application of fractions and division in problem-solving.