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

To find the arc length PQ, we can use the formula:

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

where:

( \theta = 110^{\circ} )
( r = 2.1 , \text{m} )

Therefore, the calculation becomes:

ext{Arc length} = \frac{110}{360} \times 2 \times \pi \times 2.1 \approx 4.0371 \,\text{m}

Step 2

Calculate the length of segment PQ

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Answer

To calculate the length PQ, we use the Pythagorean theorem:

PQ = \sqrt{r^2 + r^2 - 2 \cdot r \cdot r \cdot \cos(\theta)}

Inserting the known values:

PQ = \sqrt{2.1^2 + 2.1^2 - 2 \cdot 2.1 \cdot 2.1 \cdot \cos(110^{\circ})} \approx 3.4404 \,\text{m}

Step 3

Calculate the total perimeter

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Answer

The total perimeter of the shape APQBCD can be calculated by adding the lengths of sides BC, CD, and the lengths calculated earlier:

ext{Total perimeter} = (3.6 \times 2) + (8.0) + 4.0371

Evaluating that:

= 7.2 + 8.0 + 4.0371 \approx 23.7913 \,\text{m}

Rounding to one decimal place gives:

\text{Perimeter} \approx 23.8 \,\text{m}

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