Solve the problem.
-An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\theta$ is the vertex angle of the cone and $m$ is the Mach number for the speed of the plane, then $\sin \frac { \theta } { 2 } = \frac { 1 } { m }$ $( m 1 )$. Find the Mach number to the nearest tenth if $\theta = 30 ^ { \circ }$.
A) $4.5$
B) $3.7$
C) 5
D) $3.9$

Question

Solve the problem.
-An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\theta$ is the vertex angle of the cone and $m$ is the Mach number for the speed of the plane, then $\sin \frac { \theta } { 2 } = \frac { 1 } { m }$ $( m > 1 )$. Find the Mach number to the nearest tenth if $\theta = 30 ^ { \circ }$.
A) $4.5$
B) $3.7$
C) 5
D) $3.9$

Quizplus · Accepted Answer

Given $\sin \frac { \theta } { 2 } = \frac { 1 } { m }$ and $\theta = 30^\circ$, we substitute $\theta$ into the equation: $\sin \frac { 30^\circ } { 2 } = \sin 15^\circ$. Using a calculator or trigonometric tables, $\sin 15^\circ \approx 0.2588$. Thus, $\frac { 1 } { m } = 0.2588$, and solving for $m$ gives $m \approx 3.9$.

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