A circular dial on a vintage radio is rotated by \frac{5\pi}{8} radians to tune to a speci… — SAT

A circular dial on a vintage radio is rotated by $\frac{5\pi}{8}$ radians to tune to a specific station. If the dial is then rotated by an additional $45^\circ$ in the same direction, what is the total angle of rotation, in radians, of the dial?

A
$\frac{3\pi}{4}$
$\frac{7\pi}{8}$ Answer
C
$\frac{9\pi}{8}$
D
$\frac{\pi}{2}$

Answer

The correct answer is

\frac{7\pi}{8}

radians.

To find the total angle of rotation in radians, the rotation of

45^\circ

must first be converted to radians by multiplying by

\frac{\pi}{180^\circ}

, which yields

\frac{\pi}{4}

radians. Expressed with a common denominator of

8

, this is equivalent to

\frac{2\pi}{8}

radians. Adding this to the initial rotation of

\frac{5\pi}{8}

radians gives a total rotation of

\frac{5\pi}{8} + \frac{2\pi}{8} = \frac{7\pi}{8}

radians.

Step-by-Step Solution

Convert the additional rotation angle of

45^\circ

into radians.

Since

180^\circ = \pi

radians, the conversion is

45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4}

radians.

To find the total rotation angle in radians, both individual angles must be in the same unit of measure.

Add the first rotation angle of

\frac{5\pi}{8}

radians to the converted second rotation angle of

\frac{\pi}{4}

radians.

First, express

\frac{\pi}{4}

with a common denominator of

8

\frac{\pi}{4} = \frac{2\pi}{8}

. Then, add the two fractions:

\frac{5\pi}{8} + \frac{2\pi}{8} = \frac{7\pi}{8}

radians.

The total angle of rotation is the sum of the two sequential rotations in the same direction.

Key Concept

To find the sum of angles given in different units, convert the angle measured in degrees to radians using the conversion factor

\frac{\pi \text{ radians}}{180^\circ}

, and then find the sum of the two radian values using a common denominator.

Estimated Time:1m 0s

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Answer

Step-by-Step Solution

Key Concept

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