Complete the identity.
-An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\alpha$ is the vertex angle of the cone and $m$ is the Mach number for the speed of the plane, then $\sin \frac { \alpha } { 2 } = \frac { 1 } { m } ( m 1 )$. Find the Mach number to the nearest tenth if $\alpha = 90 ^ { \circ }$.
A) $1.4$
B) $1.0$
C) $1.7$
D) $2.4$

Question

Complete the identity.
-An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\alpha$ is the vertex angle of the cone and $m$ is the Mach number for the speed of the plane, then $\sin \frac { \alpha } { 2 } = \frac { 1 } { m } ( m > 1 )$. Find the Mach number to the nearest tenth if $\alpha = 90 ^ { \circ }$.
A) $1.4$
B) $1.0$
C) $1.7$
D) $2.4$

Quizplus · Accepted Answer

Given $\alpha = 90^\circ$, we substitute into the equation $\sin \frac{\alpha}{2} = \frac{1}{m}$ to find $m$. $\sin \frac{90^\circ}{2} = \sin 45^\circ = \frac{\sqrt{2}}{2}$, thus $m = \frac{1}{\sin 45^\circ} = \sqrt{2} \approx 1.4$.

Complete the Identity α\alphaα Is the Vertex Angle of the Cone And

Complete the Identity $\alpha$ Is the Vertex Angle of the Cone And