Exercise 7 Page 645

We are given that the model of a new apartment building is in the shape of the following rectangular prism.

We want to find the volume and surface area of the new building knowing that its size is $144$ times the size of the model. Let's start by finding the volume and surface area of the model. 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, $\ell =18,$ $w=10,$ and $h=28.$ We can substitute these values into the formulas to calculate the volume and surface area of the model. We will start with the volume. Next, we will calculate the surface area. Therefore, the volume of the model is $5040$ cubic inches and the surface area is $1928$ square inches. Now we can find the volume and surface area of the building using the fact that its size is $144$ times the size of the model. To do so, 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 building and its model are similar and the scale factor is $144.$ This means that we can calculate the volume and surface area of the building using these relationships. Let's do it!

Object	Volume, ${\text{in}}^3$	Surface Area, ${\text{in}}^2$
model	$5040$	$1928$
building	$5040\cdot 144^3=15049359360$	$1928\cdot 144^2=39979008$

Therefore, the volume of the building is $15049359360$ cubic inches and the surface area is $39979008$ square inches.