a spinner sector with angle 80° has arc length 14 cm. find the radius of the spinner. (\pi = 22/7)

a sector of angle 100° has an arc length of 8\pi cm. find the radius in terms of \pi, then as a decimal.

word problem: a slice of cake (sector) has angle 135° and area 47.25 cm². find the radius of the cake.

Question

a spinner sector with angle 80° has arc length 14 cm. find the radius of the spinner. (\pi = 22/7)

a sector of angle 100° has an arc length of 8\pi cm. find the radius in terms of \pi, then as a decimal.

word problem: a slice of cake (sector) has angle 135° and area 47.25 cm². find the radius of the cake.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Arc Length", "Sector Radius Calculation", "Area of a Sector" ], "new_concepts": [], "current_concepts": [ "Arc Length", "Sector Radius Calculation", "Area of a Sector" ] } ## Solve for the spinner radius $$ \begin{aligned} s &= \frac{\theta}{360^\circ} \cdot 2\pi r \ 14 &= \frac{80}{360} \cdot 2 \cdot \frac{22}{7} \cdot r \ 14 &= \frac{2}{9} \cdot \frac{44}{7} \cdot r \ 14 &= \frac{88}{63} \cdot r \ r &= 14 \cdot \frac{63}{88} = \frac{441}{44} = 10.0227\text{ cm} \end{aligned} $$ ## Solve for the second sector radius $$ \begin{aligned} s &= \frac{\theta}{360^\circ} \cdot 2\pi r \ 8\pi &= \frac{100}{360} \cdot 2\pi r \ 8\pi &= \frac{5}{18} \cdot 2\pi r \ 8\pi &= \frac{5\pi}{9} \cdot r \ r &= 8 \cdot \frac{9}{5} = \frac{72}{5} = 14.4\text{ cm} \end{aligned} $$ ## Solve for the cake radius $$ \begin{aligned} A &= \frac{\theta}{360^\circ} \cdot \pi r^2 \ 47.25 &= \frac{135}{360} \cdot \pi r^2 \ 47.25 &= \frac{3}{8} \cdot \pi r^2 \ r^2 &= \frac{47.25 \cdot 8}{3\pi} = \frac{126}{\pi} \ r &= \sqrt{\frac{126}{\pi}} \approx 6.33\text{ cm} \end{aligned} $$ ## Question 1 The radius of the spinner is $\frac{441}{44}\text{ cm}$ (or approximately $10.02\text{ cm}$). ## Question 2 The radius in terms of $\pi$ is $\frac{72}{5}\text{ cm}$ (which is independent of $\pi$ due to cancellation, or $14.4\text{ cm}$ as a decimal). ## Question 3 The radius of the cake is $\sqrt{\frac{126}{\pi}}\text{ cm}$ (or approximately $6.33\text{ cm}$). { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Sector Radius Calculation" ] }

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QUESTION IMAGE

Question

Explanation:

Solve for the spinner radius

Solve for the second sector radius

Solve for the cake radius

Answer:

Question 1

Question 2

Question 3