Question

The hands of the clock in the tower of the Houses of Parliament in London are approximately \(3 \mathrm{m}\) and \(2.5 \mathrm{m}\) in length. How fast is the distance between the tips of the hands changing at \(9.00 ?\) (Hint: Use the Law of Cosines.)

Short answer

At 9:00, the angle between the hour and minute hands is π/8 radians. Using the Law of Cosines with the given lengths, we found an equation for the distance between the tips of the hands with respect to time. After differentiating the equation with respect to time and considering that the angle does not change, we found that the rate of change of the distance between the tips of the hands is 0 meters per minute at 9:00.

Step-by-step solution

Determine the angle between the clock hands

At 9:00, the minute hand is on the 12 and the hour hand is on the 9. Every hour, the hour hand moves 1/12 of a circle, and every minute, the minute hand moves 1/60 of a circle. As 9:00 is 3/4 of an hour, the hour hand moves 3/4 * 1/12 = 1/16 of a circle. Therefore, the angle between the hour hand and the minute hand is 1/16 of a circle, or (1/16) * 2π = π/8 radians.

Apply the Law of Cosines

Let the length of the hour hand be \(a = 3 \mathrm{m}\), the length of the minute hand be \(b = 2.5 \mathrm{m}\), and the distance between the tips of the hands be \(c\). We can now use the Law of Cosines to write an equation relating the sides \(c^2 = a^2 + b^2 - 2ab\cos{\Theta}\), where \(\Theta\) is the angle between the hands, determined to be π/8 radians. Plugging in the given lengths: \(c^2 = (3 \mathrm{m})^2 + (2.5 \mathrm{m})^2 - 2(3\mathrm{m})(2.5\mathrm{m})\cos{\frac{\pi}{8}}\).

Compute the rate of change of the distance

Now we want to find the rate at which the distance between the tips of the hands is changing with respect to time. To do this, we will differentiate the equation from Step 2 with respect to time. Using implicit differentiation, we get: \(2c\frac{dc}{dt} = 0 + 0 - 2(3\mathrm{m})(2.5\mathrm{m})\frac{d}{dt}\cos{\frac{\pi}{8}}\). Since the angle \(\Theta\) does not change with time, we have \(\frac{d}{dt}\cos{\frac{\pi}{8}} = 0\), and thus: \(2c\frac{dc}{dt} = 0\). As \(c \neq 0\) (the distance between the hands can't be zero), we have: \(\frac{dc}{dt} = 0\). This means that the rate of change of the distance between the tips of the hands is 0 meters per minute at 9:00.

Key concepts

Law of Cosines

The Law of Cosines is a crucial tool in understanding non-right triangles. It's similar to the Pythagorean theorem but can be applied when we don't have a right angle in the triangle. This law helps us relate the lengths of the sides of a triangle to the cosine of one of its angles. You can think of it as a way to "measure" the unknown side when two sides and an angle are known or to find an angle when all three sides are known.
To put it simply, if you know:

• Two side lengths: let these be \(a\) and \(b\).
• The angle \(\Theta\) between them.
• The side opposite to \(\Theta\) is \(c\).

The Law of Cosines states: \[c^2 = a^2 + b^2 - 2ab\cos{\Theta}\]
By applying this formula, you can find missing side lengths or angles in various geometric problems. In the context of the clock hands, this law helps us form a relationship between the lengths of the clock hands and the angle between them at a specific time.

Implicit Differentiation

Implicit differentiation is a method used when you're dealing with equations not easily solved for one variable in terms of others. Sometimes, relationships are too complex or are naturally given in implicit form. These situations often involve equations where both variables contribute to a relationship, and solving for one isn't straightforward.
When we talk about implicit differentiation, we're essentially taking derivatives of both sides of an equation with respect to a common variable, often time \(t\). It's particularly useful for related rates problems where changes happen simultaneously to more than one variable.
Here's how it works:

• Take derivatives of each term with respect to time \(t\).
• Apply chain rules for terms that are products or compositions involving other variables.
• Solve for the desired rate of change.

In our exercise, we differentiated the distance formula derived from the Law of Cosines, assuming that all changing quantities respond to time.

Rate of Change

Understanding the rate of change is fundamental to calculus and its applications in real-world problems. It refers to how much a quantity changes with respect to another variable, often time. Think of it as the speed at which one variable reacts to changes in another.
In the context of related rates problems, you're usually looking at how one measurement changes when another influences it. At exactly 9:00 in our clock problem:

• The hour hand and minute hand form a specific angle, affecting their tips' distance.
• The rate of change in their distance tells us how quickly or slowly they're moving further apart or closer together at that moment.

The problem showed us that at 9:00, the distance between the tips wasn't changing. Therefore, the rate of change was zero. This teaches us that even within a moving system, parts may temporarily stop changing at specific times or conditions.