Slope Of A Line

To solve the problem, we need to find the distance between the points and . The formula for the distance between two points and in a Cartesian plane is given by:

Step 1: Identify the coordinates.
The given points are and .

Step 2: Substitute the coordinates into the distance formula.
Substituting the values, we get:

Step 3: Simplify the expression inside the square root.
This simplifies to:

Step 4: Calculate the square root.
We know that .

Final Answer:
The distance between the points and is .

To solve the problem, we need to determine the value of such that the slope of the line passing through the points and is 3.

1. Formula for Slope:
The slope of a line passing through two points and is given by:

In this problem, the points are and , and the slope is given as 3. Substituting these values into the formula:

2. Simplifying the Equation:
Simplify the numerator on the right-hand side:

3. Solving for :
To isolate , multiply both sides of the equation by :

Distribute the 3 on the left-hand side:

Subtract 15 from both sides:

Divide both sides by :

4. Verifying the Solution:
To ensure correctness, substitute back into the slope formula:

This confirms that the slope is indeed 3.

Final Answer:
The value of is .

To solve the problem, we need to determine the slant height of a cone given its radius and height. The formula for the slant height ( ) of a cone is derived from the Pythagorean theorem, as the slant height forms the hypotenuse of a right triangle with the radius ( ) and the height ( ) of the cone.

1. Formula for Slant Height:
The slant height of a cone is given by:

where is the radius and is the height of the cone.

2. Substituting the Given Values:
We are given that the radius and the height . Substitute these values into the formula:

Simplify the squares:

3. Approximating the Square Root:
To find the approximate value of , we can use a calculator or estimation methods. Using a calculator:

4. Final Answer:
The slant height of the cone is approximately .

Final Answer:
The slant height is .

To solve the problem, we need to find the slope of a line that makes an angle of with the positive direction of the x-axis.

1. Understanding the Relationship Between Slope and Angle:
The slope of a line is given by:


where is the angle the line makes with the positive x-axis.

2. Substituting the Given Angle:
Here,

So,

3. Evaluating the Tangent:
We know that:

and
Therefore,

Final Answer:
The slope of the line is .

To solve the problem, we need to find the angle of elevation of the top of a tower from a point on the ground at a known distance from the base of the tower.

1. Understanding the Right Triangle:
We form a right triangle where:
- Height of the tower (opposite side) = 15 m
- Distance from the base (adjacent side) = 15 m
We use the tangent function to find the angle of elevation:

2. Applying the Tangent Function:

3. Finding the Angle:

Final Answer:
The angle of elevation is