Question

an irregular park field is shown in the figure below such that cost of each square foot is 150 dollors.find the total cost of the park field

Answer 1

Answer:

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Answer 2

Given:

• Irregular field with sides 50ft, 120ft, and 40ft
• Angles formed between the sides 40° and 100°

To Find:

• Total cost of the park field

Step-by-step explanation:

1. Divide the irregular quadrilateral through the diagonal ABCD as given in the attachment.
2. Find the diagonal length using cosine law.
3. Find the area of the ΔACD using area of triangle using trigonometry formula.
4. Find ∠y using the sin rule.
5. Find ∠x using the triangle angle property.
6. Find the area of ΔABC using area of triangle using trigonometry formula.
7. Adding the area of two triangles to get the area of the quadrilateral.
8. Multiplying the cost of per square foot rate with area of the quadrilateral to find the total cost.

Solution:

Finding the length of the Diagonal AC using the cosine rule

$Formula:\\\\(AC)^{2} =(DC)^{2}+(DA)^{2}-2*DC*DA*COS( D)$$(AC)^{2} =(120)^{2}+(50)^{2}-2*120*50*COS( 40)\\\\AC)^{2} =14400+2500-2*120*50*(0.766)\\\\(AC)^{2} =16900-9192.5\\\\(AC)^{2} =7707.5\\\\(AC)=87.8ft$

Finding the area of ΔACD

$Formula:\\\\Area = \frac{1}{2} *(DC)*(DA)*SIN(D)\\\\Area=\frac{1}{2} *120*50*SIN(40)\\\\Area=1928.36\\\\Approximately,\\1928.4 square feet$,

Finding  ∠y,

$\frac{Sin(100)}{87.8} =\frac{siny}{40}$

$Sin(y)=\frac{40*Sin(100)}{87.8}$

$Sin(y)=0.449\\\\y=Sin^{-1}(0.449)\\ \\y=26 deg$

Finding ∠x,

We know the sum of all three sides of the triangle is 180°,

∠x+26°+100°=180°

∠x=180°-126°

∠x=54°

Finding the area of ΔABC,

  $Formula:\\\\Area = \frac{1}{2} *(AB)*(AC)*SIN(x)\\\\Area=\frac{1}{2} *40*87.8*SIN(54)\\\\Area=1420.633\\\\Approximately,\\1420.6square feet$

Adding both the areas of triangles,

Area of ABCD= Area of Δ ABC+ Area of ΔACD

                      $=1928.4+1420.6\\\\=3349sq feet$

Cost of park field

  $Total Cost = 150*3349\\ \\ = 502,350 dollars$

∴The total cost of the park field is 502,350 dollars