5. calculate the volume of the composite figure below to the nearest tenth. (5 marks)

6. a cylinder has a volume of $226.2\text{ cm}^3$. the radius is $3\text{ cm}$. find the height of the cylinder to the nearest tenth of a centimetre. (3 marks)

7. find the length of side ab using trigonometry. note* abc is not a right triangle. (hint. draw 2 right triangles within abc) (7 marks)

Question

5. calculate the volume of the composite figure below to the nearest tenth. (5 marks)

6. a cylinder has a volume of $226.2\text{ cm}^3$. the radius is $3\text{ cm}$. find the height of the cylinder to the nearest tenth of a centimetre. (3 marks)

7. find the length of side ab using trigonometry. note* abc is not a right triangle. (hint. draw 2 right triangles within abc) (7 marks)

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Right Triangle Trigonometry", "Geometric Decomposition", "Pythagorean Theorem" ], "new_concepts": [ "Law of Cosines", "Composite Volume" ], "current_concepts": [ "Geometric Decomposition", "Right Triangle Trigonometry", "Pythagorean Theorem", "Composite Volume", "Cylinder Volume" ] } ## Calculate composite volume Using the *Geometric Decomposition* knowledge point $$ \begin{aligned} V_{\text{cube}} &= l \cdot w \cdot h = 8 \cdot 8 \cdot 8 = 512\text{ ft}^3\ V_{\text{pyramid}} &= \frac{l \cdot w \cdot h}{3} = \frac{8 \cdot 8 \cdot 5}{3} \approx 106.7\text{ ft}^3\ V_{\text{total}} &= 512 + 106.7 = 618.7\text{ ft}^3 \end{aligned} $$ ## Solve cylinder height $$ \begin{aligned} V &= \pi r^2 h\ 226.2 &= \pi \cdot 3^2 \cdot h\ 226.2 &= 9\pi \cdot h\ h &= \frac{226.2}{9\pi} \approx 8.0\text{ cm} \end{aligned} $$ ## Decompose triangle ABC Using the *Right Triangle Trigonometry* knowledge point $$ \begin{aligned} &\text{Draw altitude } CD \text{ from } C \text{ perpendicular to } AB.\ &CD = AC \cdot \sin(46^\circ) = 16.92 \cdot \sin(46^\circ) \approx 12.17\text{ m}\ &AD = AC \cdot \cos(46^\circ) = 16.92 \cdot \cos(46^\circ) \approx 11.75\text{ m} \end{aligned} $$ ## Find remaining segment Using the *Pythagorean Theorem* knowledge point $$ \begin{aligned} &BD^2 + CD^2 = BC^2\ &BD^2 + 12.17^2 = 27.72^2\ &BD^2 \approx 768.40 - 148.11 = 620.29\ &BD \approx 24.91\text{ m} \end{aligned} $$ ## Calculate total length $$ \begin{aligned} AB &= AD + BD\ AB &\approx 11.75 + 24.91 = 36.66\text{ m} \end{aligned} $$ ## Question 5 The volume of the composite figure is 618.7 $\text{ft}^3$. ## Question 6 The height of the cylinder to the nearest tenth of a centimetre is 8.0 $\text{cm}$. ## Question 7 The length of side AB is 36.66 $\text{m}$. { "elements": [ { "type": "point", "params": [[-4, -2]], "properties": { "name": "A", "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[5, 1]], "properties": { "name": "B", "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[-1, -5]], "properties": { "name": "C", "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[-1, -1]], "properties": { "name": "D", "color": "#F2557F", "withLabel": true } }, { "type": "segment", "params": [[-4, -2], [5, 1]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[-4, -2], [-1, -5]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[-1, -5], [5, 1]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[-1, -5], [-1, -1]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1.5, "dash": 2 } } ] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Right Triangle Trigonometry" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate composite volume

Solve cylinder height

Decompose triangle ABC

Find remaining segment

Calculate total length

Answer:

Question 5

Question 6

Question 7