Problem 97 - Intermediate Algebra

Question

Two surveyors need to find the distance across a lake. They place a reference pole at point \(A\) in the diagram. Point \(B\) is 3 meters east and 1 meter north of the reference point \(A\). Point \(C\) is 19 meters east and 13 meters north of point \(A .\) Find the distance across the lake, from \(B\) to \(C\). (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The distance across the lake from B to C is 20 meters.

Step 1: Understand the Coordinate System

We are given three points:- Point \( A \) at \( (0, 0) \) because it is the reference point.- Point \( B \) at \( (3, 1) \).- Point \( C \) at \( (19, 13) \).These coordinates represent meters, with east being the x-direction and north being the y-direction.

Step 2: Use the Distance Formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\].

Step 3: Substitute the Coordinates into the Formula

Using the coordinates for points \( B \) and \( C \), substitute them into the distance formula:\[d = \sqrt{(19 - 3)^2 + (13 - 1)^2}\].

Step 4: Calculate the Differences and Squares

Calculate the differences within the formula:\(19 - 3 = 16\) and \(13 - 1 = 12\).Then square each of these differences:\(16^2 = 256\) and \(12^2 = 144\).

Step 5: Sum the Squares

Add the squares of the differences: \(256 + 144 = 400\).

Step 6: Take the Square Root

The final step is to find the square root of the sum of the squares:\(\sqrt{400} = 20\).

Step 7: Conclusion

The calculated distance from point \( B \) to point \( C \) across the lake is 20 meters.

Key Concepts

Coordinate System

The coordinate system is a fundamental concept in mathematics, especially in geometry. It allows us to represent locations in space using numerical values organized as coordinates. These coordinates are represented in usually two dimensions: the x-axis (horizontal) and the y-axis (vertical). In our problem, the points on the coordinate system are given in meters, making it easy to visualize the problem in a real-world setting.

Point \( A \) is our reference point at \( (0, 0) \), which means it is at the origin of our coordinate system. Point \( B \) is placed 3 meters east and 1 meter north, leading to its coordinates \( (3, 1) \). Similarly, point \( C \) is further out, at 19 meters east and 13 meters north, giving it coordinates \( (19, 13) \).
Understanding these coordinates helps us determine the relative position of each point, which is crucial for further calculations.

• East = x-direction
• North = y-direction
• With distances measured in meters

Geometry Problem-Solving

Geometry problem-solving often involves interpreting a diagram or a set of coordinates to derive meaningful solutions. In this exercise, we are tasked with calculating the distance across a lake between two points. This kind of problem requires logical reasoning and accuracy when substituting figures into formulas.

To solve the problem, we first needed to identify the correct geometry principles, like locating points in the coordinate system and calculating the distance between them using the distance formula. Let's recap:

1. **Identify Coordinates:** Have the exact coordinates for both points (like points \( B \) and \( C \)).
2. **Apply Formulas:** Use geometry-related formulas, like the distance formula, to find the required quantity.
3. **Accuracy:** Ensure calculations are accurate to avoid errors in the final outcome.
By following these logical steps, we methodically solve geometric problems, one step at a time.

Distance Calculation

Distance calculation is the key to solving this problem, using the distance formula from coordinate geometry. The distance formula allows us to find the length of the line segment connecting two points on the coordinate plane.

The formula is given by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].

To calculate the distance from \( B \) to \( C \), the coordinates are substituted into this formula. We find the horizontal difference between the x-values and the vertical difference between the y-values.

• Compute the x-difference: \( x_2 - x_1 = 19 - 3 = 16 \)
• Compute the y-difference: \( y_2 - y_1 = 13 - 1 = 12 \)
• Square each difference:
  \( 16^2 = 256 \) and \( 12^2 = 144 \)
• Sum the squares: \( 256 + 144 = 400 \)

Finally, take the square root of this sum to get the distance:
\( \sqrt{400} = 20 \) meters. The calculated distance shows how geometry can solve practical problems such as measuring across a lake.

Pythagorean Theorem

The Pythagorean theorem is a key component in calculating the distance between two points, especially when we treat this problem as a right triangle. The theorem states that in a right triangle, the square of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the other two sides.

In the context of our problem:

• "East" and "North" movements form a right triangle.
• The total movement east (x-difference) = 16 meters.
• The total movement north (y-difference) = 12 meters.

Using the Pythagorean theorem, we can say:
\( a^2 + b^2 = c^2 \)
where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the hypotenuse.
Here, substituting \( a = 16 \) and \( b = 12 \):
\( 16^2 + 12^2 = c^2 \)
\( 256 + 144 = c^2 \)
\( 400 = c^2 \).
Solve for \( c \) by taking the square root, we find that \( c = 20 \) meters. This confirms our previous distance calculation, showing how interconnected these geometric concepts are.