GCSE OCR Maths Pythagoras: The diagram shows some land in t

The diagram shows some land in the shape of a quadrilateral, ABCD - OCR - GCSE Maths - Question 20 - 2017 - Paper 1

Question 20

The diagram shows some land in the shape of a quadrilateral, ABCD. AB = 3 km, AD = 5 km, CD = 12 km and angle BAC = 30°. The land is sold for £10 million per squa... show full transcript

Worked Solution & Example Answer:The diagram shows some land in the shape of a quadrilateral, ABCD - OCR - GCSE Maths - Question 20 - 2017 - Paper 1

Step 1

Calculate Length AC

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Answer

To find the length of AC, we can use the Law of Cosines:

AC^2 = AB^2 + AD^2 - 2 imes AB imes AD imes ext{cos}(30^ ext{o})

Substituting the values:

AC^2 = 3^2 + 5^2 - 2 imes 3 imes 5 imes rac{ oot{3}}{2}

AC^2 = 9 + 25 - 15 imes rac{ oot{3}}{2}

Calculating gives:

AC^2 = 34 - 15 oot{3}

Thus, $AC = oot{34 - 15 oot{3}}$ km.

Step 2

Calculate Area of Triangle ABC

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Answer

The area of triangle ABC can be calculated using:

ext{Area} = rac{1}{2} imes AB imes AD imes ext{sin}(30^ ext{o})

Substituting values:

ext{Area} = rac{1}{2} imes 3 imes 5 imes rac{1}{2} = rac{15}{4} ext{ km}^2 = 3.75 ext{ km}^2.

Step 3

Calculate Area of Triangle ACD

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Answer

To find the area of triangle ACD, we can use Heron's formula. First, we need the semi-perimeter (s):

s = rac{AC + CD + AD}{2} = rac{ oot{34 - 15 oot{3}} + 12 + 5}{2}\

The area can then be calculated as:

ext{Area} = oot{s(s-AC)(s-CD)(s-AD)}.

Step 4

Calculate Total Area

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Answer

The total area of the quadrilateral ABCD is:

ext{Total Area} = ext{Area of } ABC + ext{Area of } ACD = 3.75 + ext{Area of } ACD ext{ km}^2.

Step 5

Calculate Total Cost

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Answer

The total cost of the land can be calculated using the area found: $ext{Total Cost} = ext{Total Area} imes 10 ext{ million} £/ ext{km}^2$ Substituting from the total area calculated, we find the cost in millions: $ext{Cost} = 10 imes ext{Total Area} ext{ million} £.$

The diagram shows some land in the shape of a quadrilateral, ABCD - OCR - GCSE Maths - Question 20 - 2017 - Paper 1

Worked Solution & Example Answer:The diagram shows some land in the shape of a quadrilateral, ABCD - OCR - GCSE Maths - Question 20 - 2017 - Paper 1

Calculate Length AC

Calculate Area of Triangle ABC

Calculate Area of Triangle ACD

Calculate Total Area

Calculate Total Cost

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