Question

What are the possible times when a clock shows \(35^{\circ}\) angle between two hands between \(3 \mathrm{pm}\) and \(4 \mathrm{pm}\) ? (a) 20 min \(25 \mathrm{sec}\) (b) 25 min \(20 \mathrm{sec}\) (c) \(22 \min 43 \mathrm{sec}\) (d) none of these

Short answer

Answer: (c) 22 min 43 sec

Step-by-step solution

Write the formula for the angle between the clock hands

To derive the angle \(\theta\) between the two hands of a clock knowing the time, we use the following formula: \(\theta = |30h + 0.5m - 6m|\), where \(h\) is the hour (in the range of 1 to 12) and \(m\) is the minute (in the range of 0 to 59).

Simplify the angle formula

Simplify the formula for \(\theta\) by combining the terms related to \(m\): \(\theta = |30h - 5.5m|\).

Insert the known values

Now, we will insert the known values into the formula. The angle is given as \(35^{\circ}\) and the hour is 3 (since we are looking between \(3 \mathrm{pm}\) and \(4 \mathrm{pm}\)): \(35 = |30 \times 3 - 5.5m|\).

Solve for \(m\)

Now, solve for \(m\), the number of minutes into the hour: \(35 = |90 - 5.5m|\). This equation can be split into two cases, Case 1: \(90-5.5m=35\), Solve for \(m\): \(5.5m = 55\), so \(m = 10\). Case 2: \(90-5.5m=-35\), Solve for \(m\): \(5.5m = 125\), so \(m = \frac{250}{11} \approx 22.73\).

Find the corresponding seconds

From the value of \(m\) obtained in both cases, we calculate the decimal part to find the corresponding seconds: Case 1: 10 minutes, no decimal part, so 0 seconds. Case 2: \(0.73 \times 60 \approx 43.8\) seconds, rounding gives us \(\approx 44\) seconds.

Identify the correct answer

Comparing the possible answers: (a) 20 min 25 sec (b) 25 min 20 sec (c) 22 min 43 sec (d) none of these The answer (c) 22 min 43 sec is the closest to our solution from case 2, so the possible time when a clock shows a \(35^{\circ}\) angle between its two hands between \(3 \mathrm{pm}\) and \(4 \mathrm{pm}\) is 22 min 43 sec.

Key concepts

Clock Arithmetic

Imagine a clock face, divided into 12 hours. Each hour represents an angle of 30 degrees (as one complete revolution is 360 degrees and there are 12 whole hours). Clock arithmetic is a handy way to think about how time relates to angles. For example, if it moves from 12 to 1, the angle is 30 degrees, from 1 to 2 is another 30 degrees, making 60 degrees, and so on. When solving clock angle problems, this concept is central. We need to understand the increment of degrees as each hour or minute passes.
Let's break it down for clarity:

• The minute hand moves 360 degrees in an hour, which is 6 degrees per minute since there are 60 minutes in an hour.
• The hour hand moves 30 degrees every hour but only 0.5 degrees per minute since it takes 60 minutes to move from one hour to the next.

By using these calculations, we can determine the angles formed by the hour and minute hands at any given time of day. This is the basic premise of clock arithmetic.

Angle Calculation

Calculating the angle between the clock hands for a particular time lies at the heart of clock angle problems. The angle formula \[ \theta = |30h + 0.5m - 6m| \] simplifies the process of determining the angle based on the hour and minute. Let's break this formula down to understand each component:

• 30h : This part calculates the position of the hour hand in degrees.
• 0.5m : This adds the incremental movement of the hour hand owing to the minutes passed.
• 6m : Corresponds to the degrees moved by the minute hand.

Simplifying the formula further, we have \[ \theta = |30h - 5.5m| \]. This simpler form helps calculate the absolute angle between the hour and minute hand easily, no matter what hour you’re calculating for. By substituting in values, such as an hour of 3 and solving for specific angles, you can find the precise time when the angle appears on the clock.

Trigonometry

While trigonometry may often seem complex, its application in solving clock problems is surprisingly straightforward. In the case of clock angle questions, we use basic trigonometric concepts to break down and solve for unknown angles or times.

Understanding the Context

Trigonometry deals with relationships between side lengths and angles in triangles, primarily right-angled ones. While clock arithmetic problems do not visually involve explicit triangles, the same principles can help in observing hour and minute hand movements.

Application

For clock problems, instead of dealing with trigonometric functions directly, we use the understanding of angles (derived from these functions) to approach the solution:

• We leverage the cyclic and periodic nature of trigonometric functions in angle tracking through repeated motion around the clock face.
• It aids in visualizing problems, such as noticing complementary angles depending on the quadrant location of the hands. This helps differentiate between the hour and minute positions across different times.

By applying this understanding, solving for angles or times becomes more intuitive, as students can "see" the problem not just as a number but as a geometric transition.