Problem 48 - Precalculus : Building Concepts and Connections

Question

Sarah and Joycelynn go for a hike. On the first leg, they walk 3.2 miles in the direction \(\mathrm{E} 13^{\circ} \mathrm{S}\). On the second and final leg, they walk 2.7 miles in the direction E56"S. At the end of the hike, how far are they from their starting point?

Step-by-Step Solution

Answer

The actual answer will depend on the specific trigonometric values of each component in each step. The results should be computed using a calculator.

Step 1: Break the Movements into Components

In the first movement, Sarah and Joycelynn move 3.2 miles at an angle of \(13^{\circ}\) South of East. Thus, East and South components of the first movement can be calculated using cos and sin functions respectively: \n East1 = 3.2*cos(\(13^{\circ}\)) \n South1 = 3.2*sin(\(13^{\circ}\)) \n Similarly, in the second movement, they move 2.7 miles at an angle of \(56^{\circ}\) South of East. So, East and South components of the second movement will be: \n East2 = 2.7*cos(\(56^{\circ}\)) \n South2 = 2.7*sin(\(56^{\circ}\))

Step 2: Compute total East and South displacement

Add the East and South components of both movements to determine the total distance traveled East and South: \n Total East = East1 + East2 \n Total South = South1 + South2

Step 3: Calculate the Total Displacement

Using Pythagoras' theorem, which states that the hypotenuse squared of a right triangle equals the sum of the squares of the other two sides, calculate the magnitude of the total displacement. In other words, \n Total Displacement = sqrt((Total East)^2 + (Total South)^2)

Key Concepts

Trigonometric Functions

In math, trigonometric functions help us find the relationships between angles and sides in triangles. Let's explore how they work in vector addition.

When dealing with directions such as South of East, trigonometric functions become vital. For example, when Sarah and Joycelynn hiked at an angle of \(13^{\circ}\) and \(56^{\circ}\) South of East, we used cosines and sines to determine how far they moved eastward and southward.

Here's how it works:

• Cosine \(\cos(\theta)\): This function helps calculate horizontal components—in this case, the East direction. For the first hike, it's \(3.2 \times \cos(13^{\circ})\).
• Sine \(\sin(\theta)\): This gives the vertical component—here meaning South. For the first hike, it’s \(3.2 \times \sin(13^{\circ})\).

By applying these trigonometric functions, you can effectively break down a path into its eastward and southward components. This simplifies the task of combining different directional movements into a single vector result.

Displacement Calculation

Finding displacement is like figuring out the shortest path between two points. When Sarah and Joycelynn hike, we're interested in where they end up relative to where they started.

We break each hiking leg into two components: how far east and south they go using trigonometric functions.

Once you determine these components, you add the eastward travels together and the southward travels together to find total movements in each direction.

So, the calculation steps are:

• Calculate East Components: Add the eastward steps from each hike—East1 and East2.
• Calculate South Components: Add the southward steps from each hike—South1 and South2.

With these totals, you'll know exactly how far Sarah and Joycelynn ended up east and south from their starting point, making it easy to determine their overall displacement.

Pythagorean Theorem

The Pythagorean theorem is a mathematical method that helps calculate the length of the hypotenuse—the longest side in a right triangle. This comes in handy when combining east and south movements into one straight line, or vector.

Once you have the total north and east movements, they form the two short sides of a right triangle. To figure out the hypotenuse, or straight-line distance back to the starting point, you'd use the Pythagorean theorem:

• Formula: \(\text{hypotenuse} = \sqrt{(\text{East})^2 + (\text{South})^2}\)

In simpler words, you square the total East and South distances, add them up, and then take the square root of that sum.

This calculation provides the straight-line distance, making it easy to understand how far Sarah and Joycelynn are from their starting point, regardless of the twists and turns during their hike.