Question: A zoologist tracks an animal’s path forming a right triangle with legs 9 cm and 12 cm. What is the sine of the angle opposite the 9 cm side?
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- Solution: The circumference $C = 2\pi r$, so $r = \frac{C}{2\pi} = \frac{12\pi}{2\pi} = 6$ nm.
- Question: A marine microbiome map shows a circular bacterial mat with diameter 14 cm. What is the length of the longest chord that is not a diameter?
- Solution: The longest chord in a circle is the diameter. However, the question specifies the longest chord that is not a diameter, which would approach the diameter. For a circle, the maximum chord length is the diameter, so the answer is 14 cm.
- Solution: The hypotenuse is $\sqrt{9^2 + 12^2} = 15$ cm. $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{15} = \frac{3}{5}$.
- \boxed{\dfrac{3}{5}}
- Question: A geneticist analyzes a protein structure shaped like a regular octahedron with edge length 4 nm. What is the surface area in square nanometers?
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