Exercise 30 Page 851

We are asked to find the lateral area and surface area of the given prism.

We will do these things one at a time.

Lateral Area

Let's recall the formula for the lateral area $L$ of a prism. Here, $P$ is the perimeter of the base, and $h$ the height of the prism. In this case the given prism is slanted, which means the height is measured on the outside. Examining the diagram, we can see that the height is one of the legs in a right triangle. Note that the given side is the hypotenuse and the side we want to find is opposite to the given angle. Therefore, we will use the sine ratio. In our triangle, we have that $\theta =72^{\circ },$ the hypotenuse is $18\text{ m},$ and that the opposite side to $\theta$ is $h.$ Let's substitute this information into the above formula, and solve for $h.$ We also see in the diagram that the base is a triangle and the lengths of the legs are $5,$ $13$ and $12.$ Let's add them to find its perimeter. Let's now substitute $P=30$ and $h=17.1$ in the formula for the lateral area of a prism. The lateral area of the solid is $513{\text{ m}}^2.$

Surface Area

Let's recall the formula for the surface area of a prism. Here, $L$ is the lateral area of the prism and $B$ the area of the base. We already know that $L=513{\text{ m}}^2.$ We can calculate the area of the base using the formula for area of a triangle. The area of the base is $30{\text{ m}}^2.$ Now we have enough information to find the surface area of the prism. Let's substitute $513$ and $30$ for $L$ and $B,$ respectively, into the corresponding formula. The surface area of the prism is $573{\text{ m}}^2.$