Surface Area of Big Solid: 96 square units. Relation: Area scale factor=(Linear scale factor)^2
C
cVolume of Small Solid: 6 cube units.
Volume of Big Solid: 48 cube units. Relation: Volume scale factor=(Linear scale factor)^3 Comparison: See solution.
Practice makes perfect
a To find the linear scale factor between the solids, we have to compare the length of two corresponding sides.
Examining the diagram, we see that the bigger solid has sides that are twice the length of the smaller solid. Therefore, the linear scale factor is 2.
b Note that each side is 1 unit, which means the area of each cube's sides is 1 units^2. Therefore, to find the total surface area of the solid we have to count the number of cube sides we can see from the front, back, right, left, top, and bottom view of each solid. Let's start with the smaller solid.
The number of cube sides in the smaller solid is 24. Therefore, it has an area of 24units^2. Let's also count the number of cube sides in the bigger solid.
The number of sides in the bigger solid is 96. Therefore, it has a surface area of 96 units^2. By dividing the bigger surface area with the smaller, we can find the area scale factor.
Area scale factor=96/24=4
The area scale factor between similar figures is always the square of the linear scale factor. From Part A we can recall that the linear scale factor was 2, so the area scale factor will be 2^2=4.
(Linear scale factor)^2=4
c Again, since each side is 1 unit long, each cube has a volume of (1)(1)(1)=1 unit^3. Therefore, to find the volume of each solid we have to count the number of cubes in each solid. By dividing each solid into two rectangular prisms, we can find the volume of each of these and add them to find the solid's volume. We will start with the smaller solid.
Having identified the dimensions of the two rectangular prisms, we can find the total volume of the smaller solid.
V=(1)(1)(2)+(1)(2)(2)=6 units^3
Let's also identify the dimensions of corresponding rectangular prisms in the bigger solid.
Having identified the dimensions of the two rectangular prisms, we can find the total volume of the bigger solid.
V=(2)(2)(4)+(2)(4)(4)=48 units^3
To find the volume scale factor between the solids, we have to divide the volume of the bigger solid with the volume of the smaller solid.
Volume scale factor=48/6=8
The volume scale factor between similar figures is always the cube of the linear scale factor. We again remember from Part A that the linear scale factor is 2, so 2^3=8.
(Linear scale factor)^3=8
