The diagram shows a shape APQBCD. The shape consists of a rectangle ABCD with an arc PQ on side AB and with side lengths BC = 3.6 m and CD = 8.0 m. The arc PQ is an arc of a circle with centre O and radius 2.1 m and ∠POQ = 110°. What is the perimeter of the shape APQBCD? Give your answer correct to one decimal place.

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The diagram shows a shape APQBCD - HSC - SSCE Mathematics Standard - Question 33 - 2023 - Paper 1

Question 33

The diagram shows a shape APQBCD. The shape consists of a rectangle ABCD with an arc PQ on side AB and with side lengths BC = 3.6 m and CD = 8.0 m. The arc PQ is an... show full transcript

Worked Solution & Example Answer:The diagram shows a shape APQBCD - HSC - SSCE Mathematics Standard - Question 33 - 2023 - Paper 1

Step 1

Calculate the arc length PQ

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Answer

To calculate the arc length PQ, we use the formula for the arc length of a circle:

ext{Arc length} = rac{ heta}{360} imes 2 imes rac{22}{7} imes r

Where:

$heta = 110^ ext{o}$
$r = 2.1 ext{ m}$

Now substituting in:

ext{Arc length} = rac{110}{360} imes 2 imes rac{22}{7} imes 2.1 = 4.0311 ext{ m}

Step 2

Calculate the length of line segment PQ

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Answer

The length of line segment PQ can be found using the following Pythagorean theorem:

PQ = \\sqrt{(2.1)^2 + (2.1 imes ext{cos}(110^ ext{o}))^2} \\ = \\sqrt{2.1^2 + 2.1^2 imes ext{cos}^2(110^ ext{o})} \\ = \\sqrt{2.1^2 + 2 imes 2.1 imes ext{cos}(110^ ext{o})} = 3.4404 ext{ m}

Step 3

Calculate the total perimeter

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Answer

Now, we can calculate the total perimeter of the shape APQBCD:

ext{Total Perimeter} = ext{BC} + ext{CD} + PQ + ext{Arc length PQ}

Substituting the known values:

ext{Total Perimeter} = 3.6 + 8.0 - 3.4404 + 4.0311 = 23.8 ext{ m}

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