Question

A restaurant occupying the top floor of a skyscraper rotates as diners enjoy the view. Ling and Sarah notice that they began their meal at \(7 : 00\) P.M. looking due north. At \(7 : 45\) P.M. they had rotated \(180^{\circ}\) to a view that was due south. At this rate, how many degrees will the restaurant rotate in 1 hour? A. \(180^{\circ}\) C. \(240^{\circ}\) D. \(270^{\circ}\) E. \(400^{\circ}\)

Short answer

Answer: The rotational rate of the restaurant is 240 degrees per hour.

Step-by-step solution

Identify the given information

Ling and Sarah start facing due north at 7:00 PM and rotate 180 degrees by 7:45 PM.

Calculate the time it takes to rotate 180 degrees

We need to find out how many minutes it took for the restaurant to rotate 180 degrees. Time taken = Time at 180 degrees - Initial time Time taken = 7:45 PM - 7:00 PM Time taken = 45 minutes

Find the rotational rate per minute

Divide the total degrees of rotation (180) by the minutes it took to rotate (45). Rotational rate per minute = 180 / 45 Rotational rate per minute = 4 degrees per minute

Calculate the rotation rate for 1 hour

We know the restaurant rotates 4 degrees per minute. We can multiply this by the number of minutes in an hour (60) to find the rotational rate for 1 hour. Rotational rate per hour = 4 degrees per minute * 60 minutes Rotational rate per hour = 240 degrees Thus, the restaurant rotates 240 degrees in 1 hour. So, the answer is C.

Key concepts

Rotational Motion

Rotational motion is the movement of an object around a central point. Think of it like a merry-go-round or a rotating restaurant, where every part of the object follows a circular path around the center.

In the ACT Math problem, the restaurant's top floor is the object in question, moving in a circular path as diners enjoy a panoramic view. The rate and angle of rotation help determine how far the floor will turn over a certain period. Understanding rotational motion in physics helps in predicting and calculating these movements in real-world scenarios like this one.

Degrees of Rotation

Degrees of rotation refer to the measurement of the angle through which an object turns around a fixed point. A full rotation is equal to 360 degrees, meaning if the restaurant from our ACT Math problem completes a full rotation, it would have spun a full 360 degrees.

In the problem given, diners experience a 180-degree rotation, which means they were halfway through a full circle. Accurately understanding how many degrees an object turns is crucial for solving various real-life engineering and physics problems.

Calculating Rate

Calculating rate in rotational motion terms involves how fast an object rotates, commonly expressed in degrees per minute or degrees per hour. This concept is not unlike speed in linear motion, which you might express as miles per hour.

For the rotating restaurant in our exercise, we determine how many degrees it rotates over a given period. This is akin to determining how fast a car is traveling by finding the distance it covers over a certain time. By knowing the rotational rate, we can predict where the restaurant will be facing at any given moment in time.

Time and Distance Problems

Time and distance problems in rotational motion are analogous to those in linear motion, but instead of straight-line distances, we talk about angles and degrees. The concept involves determining the position of a rotating object after a certain period based on its rotational rate.

In our ACT Math problem, understanding that time and rotational distance (degrees turned) relate to each other allowed us to calculate the rate, and in turn, know how much the restaurant will rotate in one hour. This type of problem-solving is critical in fields that involve motion, such as astronomy, mechanical engineering, and forms of transportation.