Solve using equivalent ratios.
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If we draw the diagonals of the octagonal gazebo, the 4 diagonals divide the octagon into 8 triangles.
Note that each triangle is an isosceles triangle whose equal sides are x, the radius of the circle.
The top angle of each triangle is obtained by dividing the full angle by 8.
So, each top angle =
= 45°
Now, in fig., consider one of the triangles Δ OAB. Draw an altitude OC from O to the opposite side AB.
This altitude OC bisects the top angle 45°.
Therefore, ∠ AOC = 22.5°.
Now, in Δ AOC,
So, AC = x sin 22.5°
Note that, AB = 2 AC.
Therefore, AB = 2x sin 22.5°.
Also,
So, OC = x cos 22.5°.
Area of Δ AOB =
= × (2x sin 22.5°) × (x cos 22.5°)
= (2 sin 22.5° cos 22.5°)
= sin 45°
= /
Area of the octagonal gazebo = 8 × one triangular area
= 8 × ( /
)
Area required for mulch = circular area - area of the gazebo
Now, cost per unit area = $1.50.
Hence, total cost g(m) = area × cost per unit area
Total cost g(m) = × 1.5
Hence, total cost g(m) = .
