Solve the problem.
-The mean air temperature $\mathrm { T }$, in $\mathrm { F } ^ { \circ }$, at Fairbanks, Alaska, on the nth day of the year, $1 \leq \mathrm { n } \leq 365$, is approximated by: $\mathrm { T } = 37 \sin \left( \frac { 2 \pi } { 365 } ( \mathrm { n } - 101 ) \right) + 25$. Find the temperature at Fairbanks on day 100, to the nearest tenth.
A) $24.4 ^ { \circ } \mathrm { F }$
B) $36.6 ^ { \circ } \mathrm { F }$
C) $21.4 ^ { \circ } \mathrm { F }$
D) $19.9 ^ { \circ } \mathrm { F }$

Question

Quizplus · Accepted Answer

To find the temperature on day 100, substitute $n = 100$ into the given formula: $T = 37 \sin \left( \frac { 2 \pi } { 365 } ( 100 - 101 ) \right) + 25$. This simplifies to $T = 37 \sin \left( \frac { -2 \pi } { 365 } \right) + 25$. The sine of a very small angle is approximately equal to the angle itself (in radians), so $\sin \left( \frac { -2 \pi } { 365 } \right) \approx -0.0172$. Thus, $T \approx 37(-0.0172) + 25 \approx -0.6364 + 25 \approx 24.4 ^ { \circ } \mathrm { F }$, to the nearest tenth.

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