Exercise 3 Page 644

We are given that a sink with a sliding lid is in the shape of the following rectangular prism.

We want to find the volume and surface area of a second sink knowing that it has a similar shape and is smaller by a scale factor of $\frac{1}{2}.$ Let's start by finding the volume and surface area of the larger sink. Recall that the volume and surface area of a rectangular prism with length $\ell ,$ width $w,$ and height $h$ can be calculated using the following formulas. In this case, $\ell =16,$ $w=15,$ and $h=8.$ We can substitute these values into the formulas to calculate the volume and surface area of the larger sink. We will start with the volume. Next, we will calculate the surface area. Therefore, the volume of the larger sink is $1920$ cubic inches and the surface area is $976$ square inches. Now we can find the volume and surface area of the smaller sink using the fact that it is smaller by a scale factor of $\frac{1}{2}.$ To do so, let's recall the relationships between the volumes and the surface areas of similar solids.

Volume of Similar Solids	If Solid $X$ is similar to Solid $Y$ by a scale factor, then the volume of $X$ is equal to the volume of $Y$ times the cube of the scale factor.
Surface Area of Similar Solids	If Solid $X$ is similar to Solid $Y$ by a scale factor, then the surface area of $X$ is equal to the surface area of $Y$ times the square of the scale factor.

We know that the two sinks are similar and the scale factor is $\frac{1}{2}.$ This means that we can calculate the volume and surface area of the smaller sink using these relationships. Let's do it!

Object	Volume, ${\text{in}}^3$	Surface Area, ${\text{in}}^2$
larger sink	$1920$	$976$
smaller sink	$1920{\left(\frac{1}{2}\right)}^3=240$	$976{\left(\frac{1}{2}\right)}^2=244$

Therefore, the volume of the smaller sink is $240$ cubic inches and the surface area is $244$ square inches.