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?
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- Solution: The angles are $30^\circ$, $60^\circ$, and $90^\circ$. Using the Law of Sines: $\frac{5}{\sin 30^\circ} = \frac{c}{\sin 90^\circ}$. Since $\sin 30^\circ = 0.5$ and $\sin 90^\circ = 1$, $c = \frac{5}{0.5} = 10$ km.
- Question: A geneticist models a DNA strand as a helix with a circular cross-section. If the circumference of the cross-section is $12\pi$ nm, what is the radius in nanometers?
- Solution: The circumference $C = 2\pi r$, so $r = \frac{C}{2\pi} = \frac{12\pi}{2\pi} = 6$ nm.
- 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.
- 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?
- Solution: The hypotenuse is $\sqrt{9^2 + 12^2} = 15$ cm. $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{15} = \frac{3}{5}$.
