In this section, you will apply the volume formula for a right rectangular prism in order to solve problems.

Source: Dallas Renaissance Tower 1, Andreas Praefcke, Wikimedia Commons

The Renaissance Tower in Dallas is a skyscraper that is shaped like a right rectangular prism. The base of the tower is a square, which is a special type of rectangle. The lateral faces of the tower are perpendicular to the base, and the roof is a square that is congruent to the shape of the ground floor of the building.

The square base of the Renaissance Tower has a side length of 271 feet and the building, excluding the antennae at the top, has a height of 720 feet.

Use the information provided about the Renaissance Tower to write an equation you could use to determine the volume of the interior of the building. Drag the tiles with the dimensions to the appropriate place to create an equation.

• What is the interior volume, in cubic feet, of the Renaissance Tower?

  Interactive popup. Assistance may be required.

  Need a hint?

  Use the order of operations to simplify the expression on the right side of the equation you created in the interactive.
  Interactive popup. Assistance may be required.

  Check Your Answer

  52,877,520 cubic feet
• How could you calculate the volume in cubic yards?

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  Need a hint?

  Convert each dimension from feet to yards by using the unit rate 3 yards over 1 foot 3 yards 1 foot .
  Interactive popup. Assistance may be required.

  Check Your Answer

  V = (271 ÷ 3)(271 ÷ 3)(720 ÷ 3)
  V = (901 over 3 1 3 ) (901 over 3 1 3 )(240)
  V = 1,958,4262 over 3 2 3 cubic yards

Now you will practice applying the volume formula for a right rectangular prism. To begin, you will practice writing an expression or equation from a volume formula. If you are calculating the volume, then you will have an expression that can be simplified using the order of operations. If you know the volume and are looking for one of the dimensions, then you can use inverse operations to solve the equation for the unknown dimension.

Use the interactive below to match each problem situation with an equation. It is possible that all of the equations are not necessary. To make the match, drag the equation next to the problem. Use the reference materials to help you identify the area formulas if necessary.

Use the interactive to answer the questions that follow.

• In the figure below, determine which side is the face. Explain your answer.

  

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  Check Your Answer

  Any face of the rectangular prism can be considered the base. The figure’s volume does not depend on the face selected as the base.
• How would you find the volume of a cube?

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  Check Your Answer

  A cube has a square as the base. Since a square is a rectangle and the cube is a rectangular prism, the formula for finding the volume of a rectangular prism would be used, V = Bh.

Now that you have correctly identified the equations necessary to solve each problem, use the equations to determine the solution to each of the four problems in the interactive.

Click on each problem below to check your solution.

Practice

1. The world’s deepest indoor pool is the Y-40 pool in Montegrotto Temme, Padua, Italy. It is 40 m deep. The overall length and width of the pool is 21 m x 18 m. Find the volume of the water at the deepest end if the dimensions were the same as those below.
   

   Source: World's Deepest Swimming Pool, New.com

   Interactive popup. Assistance may be required.

   Need a hint?

   Use the volume formula to find the volume of water. V = Bh, where B is the area of the base.
   Interactive popup. Assistance may be required.

   Check Your Answer

   V = 8 m x 7.5 m x 40 m
   
   V = 60 m2 x 40 m
   
   V = 2400 m³
   
   The volume of the water is 2400 m³.
2. Nancy made brownies for her friends in a 9 in x 13 in x 3 in cake pan. If the brownie batter fills the pan half-way up the side, what is the volume of the batter?

   Interactive popup. Assistance may be required.

   Need a hint?

   Use the volume formula to find the volume of the brownie batter. V = Bh, where B is the area of the base except the height will be half the height of the pan.
   Interactive popup. Assistance may be required.

   Check Your Answer

   The height of the pan will be 1.5 in since the batter fills the pan half-way.
   
   V = 9 in x 13 in x 1.5 in
   
   V = 117 in² x 1.5 in
   
   V = 175.5 in³
   
   The volume of the brownie batter is 175.5 in³.
3. Karen made her apple bread recipe at Kim’s house. Karen's recipe fills her 81 over 2 1 2 in x 41 over 2 1 2 in x 23 over 4 3 4 in bread pan to the top. Will the apple bread recipe fit in Kim’s 91 over 2 1 2 in x 31 over 2 1 2 in x 23 over 4 3 4 in. bread pan?

   Interactive popup. Assistance may be required.

   Need a hint?

   Use the volume formula to find the volume of both pans. V = Bh, where B is the area of the base. If both pans have the same volume or if the volume of Kim's pan is larger than Karen's pan, then the batter will fit in Kim’s pan.
   Interactive popup. Assistance may be required.

   Check Your Answer

   V = 81 over 2 1 2 in x 41 over 2 1 2 in x 23 over 4 3 4 in V = 12 over 2 17 2 in x 9 over 2 9 2 in x 11 over 4 11 4 in V = 153 over 4 153 4 in2 x 11 over 4 11 4 in V = 1683 over 16 1683 16 in3 = 105 3 over 16 3 16 in3	V = 91 over 2 1 2 in x 31 over 2 1 2 in x 23 over 4 3 4 in V = 19 over 2 19 2 in x 7 over 2 7 2 in x 11 over 4 11 4 in V = 133 over 4 133 4 in2 x 11 over 4 11 4 in V = 1463 over 16 1463 16 in3 = 91 7 over 16 7 16 in3
   The apple bread dough will NOT fit in Kim’s pan.