Unit 4 Lesson 15: Volume of Prisms

A pool in the shape of a rectangular prism is being filled with water. The length and width of the pool is 24 feet and 15 feet. If the height of the water in the pool is  feet, what is the volume of the water in cubic feet? $1\ \frac{1}{3}$  

• 6,360 squared feet. ( $24\cdot15=6360$ ) 

• 480 cubic feet. ( $424\cdot15=360$ , and $360\cdot\frac{4}{3}=480$ .)

• 3,180 squared feet. ( $\frac{1}{2}\cdot24\cdot15=3,180$ ) 

A rectangular prism measures  $2\ \frac{2}{5}$  inches by   $3\ \frac{1}{5}$  inches by 2 inches.
Priya said, “It takes more cubes with edge length  $\frac{2}{5}$  inch than cubes with edge length   $\frac{1}{5}$ inch to pack the prism.” Do you agree with Priya’s statement? Explain your reasoning.

• Agree. Cubes with side lengths  $\frac{2}{5}$   inch are smaller than cubes with side lengths  $\frac{1}{5}$  inch, so it would take more of the former to pack the same prism.

• Disagree.

• Disagree. Cubes with side lengths  $\frac{2}{5}$   inch are larger than cubes with side lengths  $\frac{1}{5}$  inch, so it would take fewer of the former to pack the same prism.

A rectangular prism measures $2\ \frac{2}{5}$  inches by  $3\ \frac{1}{5}$ inches by 2 inches.

How many cubes with edge length  $\frac{1}{5}$  inch fit in the prism? Show your reasoning.

• 38 cubes. My reasoning:  $2\ \frac{2}{5}\div\frac{1}{5}=12$  ,  $3\ \frac{1}{5}\div\frac{1}{5}=16$  , and  $2\div\frac{1}{5}=10$ .  We can fit 12 cubes along the length of the prism, 16 cubes along the width, and 10 cubes along the height, so the number of cubes is:  $12+16+10=38$  . 

• 1,920 cubes. My reasoning:  $2\ \frac{2}{5}\div\frac{1}{5}=12$ , $3\ \frac{1}{5}\div\frac{1}{5}=16$  , and  $2\div\frac{1}{5}=10$ . We can fit 12 cubes along the length of the prism, 16 cubes along the width, and 10 cubes along the height, so the number of cubes is:  $12\cdot16\cdot10=1,920$ .

• 192 cubes. My reasoning:  $2\ \frac{2}{5}\div\frac{1}{5}=12$ ,  $3\ \frac{1}{5}\div\frac{1}{5}=16$ . We can fit 12 cubes along the length of the prism and 16 cubes along the width, so the number of cubes is:  $12\cdot16=192$ .

Here is a right triangle. What is its area?

• $\frac{3}{8}cm^2$ . The area of a triangle is found with $\frac{1}{2}\cdot base\cdot height$ . We can use the two perpendicular sides as the base and the height $\frac{1}{2}\cdot\frac{3}{4}\cdot1=\frac{3}{8}$ .

• $\frac{3}{4}cm^2$ . The area of a triangle is found with $base\cdot height$ . We can use the two perpendicular sides as the base and the height $\frac{3}{4}\cdot1=\frac{3}{4}$ .

• I cannot solve this because the height is not provided.

What is the height $h$  for the base that is  $\frac{5}{4}$  units long if the area is  $\frac{3}{8}cm^2$ ? Show your reasoning.

•   $\frac{3}{5}cm$ . The area of the triangle is  $\frac{3}{8}cm^2$   and we can also write the area using the  $\frac{5}{4}$  side and  $h$ . $\frac{1}{2}\cdot\frac{5}{4}h=\frac{3}{8}$  , so  $\frac{5}{8}h=\frac{3}{8}$ . To find what we could multiply by $\frac{5}{8}$  to get $\frac{3}{8}$  , we can divide  $\frac{3}{8}\div\frac{5}{8}$ , which is  $\frac{3}{8}\cdot\frac{8}{5}$  , which is  $\frac{3}{5}$ .

• 1cm. The height is 1cm because that is the number to the right of the base.

•  $\frac{15}{64}cm$  . The area of the triangle is  $\frac{3}{8}cm^2$  and we can also write the area using the  $\frac{5}{4}$  side and  $h$ .  $\frac{5}{8}\cdot\frac{3}{8}=h$ . 

To give their animals essential minerals and nutrients, farmers and ranchers often have a block of salt—called “salt lick”—available for their animals to lick.
A rancher is ordering a box of cube-shaped salt licks. The edge lengths of each salt lick are $\frac{5}{12}$ foot. Is the volume of one salt lick greater or less than 1 cubic foot? Explain your reasoning.

• Greater than 1 cubic foot. A cube with edge length 1 foot has a volume of 1 cubic foot. A salt-lick cube has edge length $\frac{5}{12}$  foot, which is less than 1 foot, so its volume ( $\frac{5}{12}\cdot \frac{5}{12}\cdot \frac{5}{12}$ ) is greater than 1 cubic foot. 

• 1 cubic foot. A cube with edge length 1 foot has a volume of 1 cubic foot. A salt-lick cube has edge length $\frac{5}{12}$  foot, which is less than 1 foot, so its volume ( $\frac{5}{12}\cdot \frac{5}{12}\cdot \frac{5}{12}$ ) is equal to 1 cubic foot. 

• Less than 1 cubic foot. A cube with edge length 1 foot has a volume of 1 cubic foot. A salt-lick cube has edge length $\frac{5}{12}$  foot, which is less than 1 foot, so its volume ( $\frac{5}{12}\cdot\frac{5}{12}\cdot\frac{5}{12}$ ) is less than 1 cubic foot. 

• I cannot answer this. There is not enough information.

To give their animals essential minerals and nutrients, farmers and ranchers often have a block of salt—called “salt lick”—available for their animals to lick.

The box that contains the salt lick is $1\ \frac{1}{4}$ feet by  $1\ \frac{2}{3}$ feet by $\frac{5}{6}$ feet. How many cubes of salt lick fit in the box? Explain your reasoning.

• 24 cubes. The length of the box can fit  $\frac{5}{4}\div\frac{5}{12}$  or 3 cubes. The width of the box can fit  $\frac{5}{3}\div\frac{5}{12}$  or 4 cubes. The height of the box can fit   $\frac{5}{6}\div\frac{5}{12}$ or 2 cubes. The box can fit ( $3\cdot4\cdot2$ )  or 24 cubes.

• 12 cubes. The length of the box can fit  $\frac{5}{4}\div\frac{5}{12}$ or 3 cubes. The width of the box can fit  $\frac{5}{3}\div\frac{5}{12}$ or 4 cubes. The box can fit ( $3\cdot4$ ).   

• 8 cubes. The width of the box can fit  $\frac{5}{3}÷\frac{5}{12}$  or 4 cubes. The height of the box can fit   $\frac{5}{6}÷\frac{5}{12}$ or 2 cubes. The box can fit ( $4\cdot2$ )  or 8 cubes.

• 6 cubes. The length of the box can fit  $\frac{5}{4}÷\frac{5}{12}$  or 3 cubes.  The height of the box can fit   $\frac{5}{6}÷\frac{5}{12}$ or 2 cubes. The box can fit ( $3\cdot2$ )  or 6 cubes.

 How many groups of $\frac{1}{3}$  inch are in $\frac{3}{4}$  inch?

•  $2\ \frac{1}{4}\$ .  To find "how many groups," compute  $\frac{3}{4}\div\frac{1}{3}$ , which is $\frac{3}{4}\cdot\frac{3}{1}$  , which is   $\frac{9}{4}$ or $2\ \frac{1}{4}$ . 

•  $\frac{1}{4}$ .  To find "how many groups," compute  $\frac{3}{4}÷\frac{1}{3}$ , which is $\frac{3}{4}\cdot\frac{1}{3}$  , which is   $\frac{3}{12}$ or $\frac{1}{4}$ . 

•  $\frac{4}{7}$ . To find "how many groups," compute  $\frac{3}{4}+\frac{1}{3}$ , which is  $\frac{4}{7}$ .   

•  $\frac{4}{9}$  .  To find "how many groups," compute  $\frac{1}{3}\div\frac{3}{4}$ , which is $\frac{1}{3}\cdot\frac{4}{3}$  , which is   $\frac{4}{9}$ .

How many inches are in $1\frac{2}{5}$ groups of  $1\ \frac{2}{3}$ inches?

•  $\frac{21}{25}$ . To find "how many inches," compute  $1\ \frac{2}{5}\div1\ \frac{2}{3}$ , which is $\frac{7}{5}\div\frac{5}{3}$  , which is $\frac{7}{5}\cdot\frac{3}{5}$ , which is  $\frac{21}{25}$ .

•  $1\ \frac{4}{7}$ . To find "how many inches," compute  $1\ \frac{2}{3}\div1\ \frac{2}{5}$ , which is $\frac{5}{3}\div\frac{7}{5}$  , which is $\frac{5}{3}\cdot\frac{5}{7}$ , which is $\frac{25}{21}$ or  $1\ \frac{4}{7}$ .

•  $2\ \frac{1}{3}$ . To find "how many inches," compute  $1\ \frac{2}{5}\cdot1\ \frac{2}{3}$ , which is $\frac{7}{5}\cdot\frac{5}{3}$  , which is $\frac{7}{3}$ or  $2\ \frac{1}{3}$ .

Here is a table that shows the ratio of flour to water in an art paste. Complete the table with values in equivalent ratios.

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