Solve the problem.
-Suppose that a radio signal transmission pattern can be modeled by $\mathrm { r } ^ { 2 } = \mathrm { k } \cos 2 \theta$, for $0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$, where the units are in miles. Describe the influence of $k$ on the pattern. Assume that the positive $x$-axis points east.

A) With increasing $k$, the transmission becomes more strongly directed toward the north and less strongly directed to the east.
B) With increasing $\mathrm { k }$, the distance of transmission becomes smaller in the east and west directions.
C) With increasing $k$, the distance of transmission becomes greater in the east and west directions.
D) With increasing $\mathrm { k }$, the transmission becomes more strongly directed toward the east and less strongly directed to the west.

Question

Solve the problem.
-Suppose that a radio signal transmission pattern can be modeled by $\mathrm { r } ^ { 2 } = \mathrm { k } \cos 2 \theta$, for $0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$, where the units are in miles. Describe the influence of $k$ on the pattern. Assume that the positive $x$-axis points east.

A) With increasing $k$, the transmission becomes more strongly directed toward the north and less strongly directed to the east.
B) With increasing $\mathrm { k }$, the distance of transmission becomes smaller in the east and west directions.
C) With increasing $k$, the distance of transmission becomes greater in the east and west directions.
D) With increasing $\mathrm { k }$, the transmission becomes more strongly directed toward the east and less strongly directed to the west.

Quizplus · Accepted Answer

The equation $ r^2 = k \cos 2\theta $ represents a polar equation where $k$ influences the radius $r$ of the transmission pattern. As $k$ increases, the value of $r^2$ increases for given values of $\theta$, meaning the distance of transmission becomes greater. Specifically, when $\theta = 0^\circ$ or $180^\circ$ (east or west directions), $\cos 2\theta = \cos 0 = 1$, maximizing $r^2 = k$, thus increasing $k$ increases the transmission distance in these directions.

Solve the Problem r2=kcos⁡2θ\mathrm { r } ^ { 2 } = \mathrm { k } \cos 2 \thetar2=kcos2θ

Solve the Problem $\mathrm { r } ^ { 2 } = \mathrm { k } \cos 2 \theta$