Exercise 89 Page 635

The volume of the fish tank is the area of its base multiplied by the height. Since it is a regular octagon, we can find the measure of each interior angle by substituting n=8 into the formula 180^(∘)(n-2)n and then simplifying. Each interior angle is 135^(∘). Let's illustrate this in a diagram.

To calculate the area of the base, we can draw diagonals between opposite vertices and create 8 congruent isosceles triangles. If we obtain the area of one triangle we can find the area of the base by multiplying this value by 8.

By using the tangent ratio we can calculate the height of the triangle.

tan θ =Opposite/Adjacent

SubstituteValues

Substitute values

tan 67.5^(∘) =h/0.4

▼

Solve for h

MultEqn

LHS * 0.4=RHS* 0.4

0.4tan 67.5^(∘) = h

RearrangeEqn

Rearrange equation

h = 0.4tan 67.5^(∘)

UseCalc

Use a calculator

h = 0.9567...

RoundDec

Round to 2 decimal place(s)

h ≈ 0.966

The height of the triangle is about 0.966 meters. With this information we can find the area of the triangle and finally, if we multiply this by 8, the area of the base. A=(1/2(0.8)(0.966))8 ≈ 3.09 m^2 The base has an area of about 3.09 m^2. To find the prism's volume, we multiply the area of the base by the prism's height. V=3.09(2)≈ 6.18m^3 To find the surface area of the fish tank we have to add the area of the two bases with the area of the sides. The area of one base is 3.09 m^2, which means the area of both bases must be 3.09 * 2 = 26.18 m^2. Each side is rectangular with a width of 0.8 meters and a length of 2 meters. Since there are 8 sides, we can find the total area of the sides by multiplying the area of one side by 8. A=(2)(0.8)(8)=12.8 m^2 Finally, we will add all of the external faces to get the total surface area of the fish tank. 6.18+12.8=18.98 m^2