10 the cross-section of a piece of conduit is an annulus. it has an outer diameter of $60\text{ mm}$ and an inner diameter of $50\text{ mm}$.
a find the area of the cross-section to the nearest square millimetre.
b what is the increase in cross-sectional area if the outer diameter is increased to $65\text{ mm}$? answer correct to the nearest square millimetre.

11 an annulus with an inner diameter of $4.2\text{ metres}$ and an outer diameter of $10\text{ metres}$ is divided into two equal parts. what is the area of each part? answer correct to three significant figures.

12 the area of a circular pond is $5.3\text{ square metres}$. what is the radius of the pond? answer correct to one decimal place.

13 a traffic roundabout has a circular garden in the centre and two lanes for traffic encircling the garden. the diameter of the garden is $16\text{ metres}$ and each lane is $3\text{ metres}$ wide. each lane is to be resurfaced. calculate the area to be resurfaced. answer in square metres to the nearest whole number.

Question

10 the cross-section of a piece of conduit is an annulus. it has an outer diameter of $60\text{ mm}$ and an inner diameter of $50\text{ mm}$.
a find the area of the cross-section to the nearest square millimetre.
b what is the increase in cross-sectional area if the outer diameter is increased to $65\text{ mm}$? answer correct to the nearest square millimetre.

11 an annulus with an inner diameter of $4.2\text{ metres}$ and an outer diameter of $10\text{ metres}$ is divided into two equal parts. what is the area of each part? answer correct to three significant figures.

12 the area of a circular pond is $5.3\text{ square metres}$. what is the radius of the pond? answer correct to one decimal place.

13 a traffic roundabout has a circular garden in the centre and two lanes for traffic encircling the garden. the diameter of the garden is $16\text{ metres}$ and each lane is $3\text{ metres}$ wide. each lane is to be resurfaced. calculate the area to be resurfaced. answer in square metres to the nearest whole number.

Accepted Answer

{ "quality": "clear", "question_count": 5, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Annulus Area", "Circle Area" ], "new_concepts": [], "current_concepts": [ "Annulus Area", "Circle Area" ] } ## Calculate the initial cross-sectional area Using the *Annulus Area* knowledge point $$ \begin{aligned} &R_1 = \frac{60}{2} = 30\text{ mm},\quad r_1 = \frac{50}{2} = 25\text{ mm}\ &A_1 = \pi (R_1^2 - r_1^2) = \pi (30^2 - 25^2) = 275\pi \approx 863.938\text{ mm}^2\ &A_1 \approx 864\text{ mm}^2 \end{aligned} $$ ## Calculate the new area and the increase Using the *Annulus Area* knowledge point $$ \begin{aligned} &R_2 = \frac{65}{2} = 32.5\text{ mm},\quad r_1 = 25\text{ mm}\ &A_2 = \pi (R_2^2 - r_1^2) = \pi (32.5^2 - 25^2) = 431.25\pi \approx 1354.757\text{ mm}^2\ &\Delta A = A_2 - A_1 = 431.25\pi - 275\pi = 156.25\pi \approx 490.819\text{ mm}^2\ &\Delta A \approx 491\text{ mm}^2 \end{aligned} $$ ## Calculate the area of half the annulus Using the *Annulus Area* knowledge point $$ \begin{aligned} &R = \frac{10}{2} = 5\text{ m},\quad r = \frac{4.2}{2} = 2.1\text{ m}\ &A_{\text{half}} = \frac{1}{2} \pi (R^2 - r^2) = \frac{1}{2} \pi (5^2 - 2.1^2) = 10.295\pi \approx 32.342\text{ m}^2\ &A_{\text{half}} \approx 32.3\text{ m}^2 \end{aligned} $$ ## Find the radius of the circular pond Using the *Circle Area* knowledge point $$ \begin{aligned} &A = \pi r^2 = 5.3\text{ m}^2\ &r^2 = \frac{5.3}{\pi} \approx 1.687\ &r = \sqrt{\frac{5.3}{\pi}} \approx 1.2988\text{ m}\ &r \approx 1.3\text{ m} \end{aligned} $$ ## Calculate the resurfacing area of the lanes Using the *Annulus Area* and *Circle Area* knowledge points $$ \begin{aligned} &r_{\text{garden}} = \frac{16}{2} = 8\text{ m}\ &R_{\text{outer}} = 8 + 3 + 3 = 14\text{ m}\ &A_{\text{lanes}} = \pi (R_{\text{outer}}^2 - r_{\text{garden}}^2) = \pi (14^2 - 8^2) = 132\pi \approx 414.69\text{ m}^2\ &A_{\text{lanes}} \approx 415\text{ m}^2 \end{aligned} $$ ## Question 10 a $864\text{ mm}^2$ b $491\text{ mm}^2$ ## Question 11 $32.3\text{ m}^2$ ## Question 12 $1.3\text{ m}$ ## Question 13 $415\text{ m}^2$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Annulus Area" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the initial cross-sectional area

Calculate the new area and the increase

Calculate the area of half the annulus

Find the radius of the circular pond

Calculate the resurfacing area of the lanes

Answer:

Question 10

Question 11

Question 12

Question 13