Exercise 31 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

The lateral area L of a prism is the sum of the areas of the lateral faces. Let's recall the formula for the lateral area of a prism. L=PhIn this formula, P is the perimeter of the base and h the height of the prism. In the diagram, we see that the height is one of the legs in a right triangle.

Note that the given side is the opposite to the given angle, and the side we want to find is the hypotenuse. Therefore, we will use the sine ratio. sin θ = opposite/hypotenuse In our triangle, we have that θ = 59^(∘), the opposite side to θ is 18 cm. Let x be the hypotenuse of the triangle. We will substitute this information into the above formula, and solve for x.

The lateral area of the oblique prism contains two parallelograms and two rectangles. We can find the lateral area by adding the areas of each lateral face. We will use the formulas for the area of a parallelogram and the area for a rectangle.

The lateral area of the solid is 1392.0 cm^2.

Surface Area

Let's now recall the formula for the surface area of a prism S. S=L+2B Here, L is the lateral area and B is the area of the base. Notice that the base is a rectangle with length 20 and width 16. Therefore, to find its area, we will multiply these two numbers. B=( 20)( 16) ⇔ B=320 cm^2 The area of the base is 320cm^2. Now we have enough information to find the surface area of the prism. Let's substitute L with 1392 and B with 320 into the formula.

The surface area of the prism is 2032cm^2.