Skill: Determine the effect of scale factors on area and volume

A scale factor $k$ changes measurements in different ways:

• Lengths are multiplied by $k$.
• Areas are multiplied by $(k^2)$.
• Volumes are multiplied by $(k^3)$.

The exponent tells you the type of measurement: $1$ for length, $2$ for area, and $3$ for volume.

Example: A rectangular prism has dimensions $(2\text{ cm})$, $(3\text{ cm})$, and $(4\text{ cm})$. It is enlarged by a scale factor of $(1.5)$. Find its new dimensions, surface area, and volume.

Step 1: Find the new dimensions.

Multiply each length by $(1.5)$:

$$2(1.5)=3\text{ cm}$$ $$3(1.5)=4.5\text{ cm}$$ $$4(1.5)=6\text{ cm}$$

The new dimensions are $(3\text{ cm})$, $(4.5\text{ cm})$, and $(6\text{ cm})$.

Step 2: Find the new surface area using the area scale factor.

Surface area is an area, so multiply the original surface area by $(1.5^2)$.

First find the original surface area:

$$SA=2(lw+lh+wh)$$ $$SA=2(2\cdot3+2\cdot4+3\cdot4)$$ $$SA=2(6+8+12)=52\text{ cm}^2$$

Now apply the area scale factor:

$$1.5^2=2.25$$ $$\text{New surface area}=52(2.25)=117\text{ cm}^2$$

Step 3: Find the new volume using the volume scale factor.

Volume is a volume, so multiply the original volume by $(1.5^3)$.

$$V=2\cdot3\cdot4=24\text{ cm}^3$$ $$1.5^3=3.375$$ $$\text{New volume}=24(3.375)=81\text{ cm}^3$$

Therefore, the enlarged prism has:

$$\boxed{\text{Dimensions: }3\text{ cm}\times4.5\text{ cm}\times6\text{ cm}}$$ $$\boxed{\text{Surface area: }117\text{ cm}^2}$$ $$\boxed{\text{Volume: }81\text{ cm}^3}$$

Remember: use $k$ for lengths, $(k^2)$ for areas, and $(k^3)$ for volumes.