a The volume of any prism is the base area multiplied by the prism's height. From the exercise we know that Paul's tower is 6 blocks high. Let's have a look at the base area from a top-side view and count the number of squares we have.
As we can see the base area consists of 7 squares, which means we have a base area of 7 square units. If we multiply this by the tower's height, which is 6 units, we get the volume of the tower.
7(6)=42 units^3
b The surface area is the combined area of the tower's lateral faces and base areas. From the exercise we know that the base area is 7 square units. Since we have two bases, top and bottom, we get a total area of 14 square units. To calculate the lateral faces we measure the length of the base area's sides.
Now we can calculate the lateral face's area by adding all of the side lengths and multiplying this by the height of the tower.
6(2( 2)+7( 1)+ 3)=84
Finally, we add the sum of the lateral faces with the sum of the base areas.
84+14=98 units^2
c As in Part A and B, we will find the volume of the prisms by first identifying the base area and then multiplying this by their respective heights. Let's have a look at the top side view of each prism and identify their areas.
When we have identified the base area of each prism, we multiply by the height to determine the volume. Examining the diagram, we see that the first and third prism have a height of 5 units. The second prism has a height of 4 units. Now we can determine the volume of each prism.
Calculating Volumes
[-1em]
(1):& 4(5) -10pt&&=20 units^3
(2):& 6(4) -10pt&&=24 units^3
(3):& 12(5) -10pt&&=60 units^3
Now we will find the surface area. This is the sum of the base area and lateral faces. The lateral face's area is the perimeter of each prism multiplied by the height.
Calculating Surface Areas
[-1em]
(1):& 10(5)+ 2(4) -10pt&&=58 units^2
(2):& 12(4)+ 2(6) -10pt&&=60 units^2
(3):& 14(5)+ 2(12) -10pt&&=94 units^2
