The following table lists the distance and time took by Volcanic Smog to spread from Taal Volcano up to the nearby areas. The table also shows the bearings of the smog. A bearing is the angle formed with the vertical by a line segment connecting Batangas and Volcanic Smog from Taal Volcano (Batangas): Point, Distance from
Taytay (km), Bearing, Time, Date:
A (Batangas City) 105.2km, 94° 10:30 AM Aug 20
B (Lipa) 81.4km, 88° 12:30 PM Aug 20
C (Tanauan) 64.7km, 74° 1:30 PM Aug 20
D (Calamba) 54.5km, 72° 2:20 PM Aug 20
E (Cabuyao) 47.3km, 60° 3:20 PM Aug 20

Find the slope of the line from point A to E

To find the slope of the line from point A (Batangas City) to point E (Cabuyao), we need to calculate the slope using the general formula for the slope between two points:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of points A and E, respectively. In this case, we can calculate the coordinates based on the distances and bearings provided in the table.

For each point, we will first convert the bearings into Cartesian coordinates using the formulas:

\[
x = r \cdot \sin(\theta)
\]
\[
y = r \cdot \cos(\theta)
\]

Where \(r\) is the distance from the volcano, and \(\theta\) is the bearing (in degrees) measured clockwise from the north (or vertical).

### Coordinates of Point A (Batangas City):
- Distance = 105.2 km
- Bearing = 94° (which is \(90° + 4°\) from east)

\[
x_A = 105.2 \cdot \sin(94°)
\]
\[
y_A = 105.2 \cdot \cos(94°)
\]

### Coordinates of Point E (Cabuyao):
- Distance = 47.3 km
- Bearing = 60°

\[
x_E = 47.3 \cdot \sin(60°)
\]
\[
y_E = 47.3 \cdot \cos(60°)
\]

Once we have the coordinates \((x_A, y_A)\) and \((x_E, y_E)\), we can use the slope formula to find the slope of the line from point A to point E. Let's calculate that.The Cartesian coordinates for the two points are as follows:
- Point A (Batangas City): \( (104.94, -7.34) \)
- Point E (Cabuyao): \( (40.96, 23.65) \)

Using these coordinates, the slope of the line from point A to point E is approximately:

\[
m \approx -0.484
\]

This means that the line slopes downward as we move from Batangas City to Cabuyao.

Would you like further details, or do you have any questions?

Here are 5 related questions:
1. How does the bearing affect the calculation of slope between two points?
2. Can the slope be used to estimate the rate of spread of the volcanic smog?
3. What does a negative slope signify in this geographical context?
4. How would the slope change if the bearing angles were larger?
5. Could this method be applied to other natural phenomena, like wind direction?

**Tip:** When dealing with bearings, always ensure you convert degrees to radians when using trigonometric functions.

The following table lists the distance and time took by Volcanic Smog to spread from Taal Volcano up to the nearby areas. The table also shows the bearings of the smog. A bearing is the angle formed with the vertical by a line segment connecting Batangas and Volcanic Smog from Taal Volcano (Batangas): Point, Distance from Taytay (km), Bearing, Time, Date: A (Batangas City) 105.2km, 94° 10:30 AM Aug 20 B (Lipa) 81.4km, 88° 12:30 PM Aug 20 C (Tanauan) 64.7km, 74° 1:30 PM Aug 20 D (Calamba) 54.5km, 72° 2:20 PM Aug 20 E (Cabuyao) 47.3km, 60° 3:20 PM Aug 20. Find the slope of the line from point A to E.

Learn how to calculate the slope of the line between two points based on bearings and distances from Taal Volcano. This problem involves converting bearings into Cartesian coordinates using trigonometric functions and applying the slope formula. Suitable for students in Grades 9-12.

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