Lesson 14

1. The area of Triangle B is 8 square units. Find the length of $b$. Show your reasoning.

   

   

2. The area of Triangle C is $\frac{54}{5}$ square units. What is the length of $h$? Show your reasoning.

   

   

Use cubes or the applet to help you answer the following questions. 

1. Here is a drawing of a cube with edge lengths of 1 inch.

   1. How many cubes with edge lengths of $\frac12$ inch are needed to fill this cube?
   2. What is the volume, in cubic inches, of a cube with edge lengths of $\frac12$ inch? Explain or show your reasoning.

   

   

2. Four cubes are piled in a single stack to make a prism. Each cube has an edge length of $\frac12$ inch. Sketch the prism, and find its volume in cubic inches.

3. Use cubes with an edge length of $\frac12$ inch to build prisms with the lengths, widths, and heights shown in the table.

   1. For each prism, record in the table how many $\frac12$-inch cubes can be packed into the prism and the volume of the prism.

      prism length (in)	prism width (in)	prism height (in)	number of $\frac12$-inch cubes in prism	volume of prism (in3)
      $\frac12$	$\frac12$	$\frac12$
      1	1	$\frac12$
      2	1	$\frac12$
      2	2	1
      4	2	$\frac32$
      5	4	2
      5	4	$2\frac12$

   2. Examine the values in the table. What do you notice about the relationship between the edge lengths of each prism and its volume?
4. What is the volume of a rectangular prism that is $1\frac12$ inches by $2\frac14$ inches by 4 inches? Show your reasoning.

To find the volume of a rectangular prism with fractional edge lengths, we can think of it as being built of cubes that have a unit fraction for their edge length. For instance, if we build a prism that is $\frac12$-inch tall, $\frac32$-inch wide, and 4 inches long using cubes with a $\frac12$-inch edge length, we would have:

• A height of 1 cube, because $1 \boldcdot \frac 12 = \frac12$.
• A width of 3 cubes, because $3 \boldcdot \frac 12 = \frac32$.
• A length of 8 cubes, because $8 \boldcdot \frac 12 = 4$.

The volume of the prism would be $1 \boldcdot 3 \boldcdot 8$, or 24 cubic units. How do we find its volume in cubic inches? We know that each cube with a $\frac12$-inch edge length has a volume of $\frac 18$ cubic inch, because $\frac 12 \boldcdot \frac 12 \boldcdot \frac 12 = \frac18$. Since the prism is built using 24 of these cubes, its volume, in cubic inches, would then be $24 \boldcdot \frac 18$, or 3 cubic inches.

The volume of the prism, in cubic inches, can also be found by multiplying the fractional edge lengths in inches: ​​​​​​$\frac 12 \boldcdot  \frac 32 \boldcdot 4 = 3$