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Let x be the length of each side of the nonagon. We then split up the nonagon into 9 congruent, isosceles triangles, each with base = x and height = 12. Then the area of each triangle is 1/2 • x • 12 = 6x, so the total area of the nonagon will be 9 • 6x = 54x.
To find x, we can use some facts from geometry and trigonometry.
• In any polygon, the sum of the measures of the exterior angles is 360°. So each of these exterior angles will measure 360°/9 = 40°.
• Exterior angles are supplementary to the interior angles. So each interior angle will measure 180° - 40° = 140°.
• Each of the 9 triangles are isosceles with base angles measuring half the interior angles of the nonagon, 140°/2 = 70°.
• Cut the triangle in half along the labeled inradius of the nonagon, which has length 12. In the resulting right triangle, we have
tan(70°) = 12 / (x/2)
and solving for x gives
tan(70°) = 24/x
x = 24/tan(70°)
x = 24 cot(70°) ≈ 8.7
Then the total area of the nonagon is
54x = 54 • 24 cot(70°) ≈ 471.7
Answer:
C(x) =
Step-by-step explanation:
Data provided in the question:
Area of the field = 3600 square feet
Fencing charges = $3.50 per foot
Let the side along the river be 'x' feet and the other side of the 'B'
now,
Area of rectangle = Bx = 3600 square feet
or
B = feet
and total length to be fenced = x + 2B
therefore,
Total cost of fencing = Fencing charges × total length to be fenced
or
Total cost of fencing = $3.50 × ( x + 2B )
substituting the value of B from (1)
Total cost of fencing, C(x) =
or
C(x) =