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 Advanced - Question 16 - 2023 - Paper 1

Question 16

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 Advanced - Question 16 - 2023 - Paper 1

Step 1

Calculate the arc length PQ

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Answer

The length of the arc PQ can be calculated using the formula:

ext{Arc Length} = rac{ heta}{360} imes 2 imes ext{r}

where $heta$ is the angle in degrees and r is the radius. For our shape, $heta = 110^ ext{o}$ and $r = 2.1 ext{ m}$ :

ext{Arc Length} = rac{110}{360} imes 2 imes 3.14 imes 2.1 \\ = 4.0317 ext{ m (approximately)}

Step 2

Calculate the length of segment PQ

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Answer

The length of segment PQ can be calculated using the formula derived from the Pythagorean theorem:

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

Step 3

Calculate the total perimeter

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Answer

The total perimeter of shape APQBCD can be calculated as follows:

ext{Total Perimeter} = (3.6 imes 2) + (8.0) + (4.0317) \\ = 23.8 ext{ m (correct to one decimal place)}

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