a tower 150 meter is situated at the top of the hill.At a point 650 m down the hill.The angle between the surface of the hill and the line of sight to the of the tower is 12degrees 30 '.Find the inclination of the hill to a horizontal plane.
http://www.jiskha.com/display.cgi?id=1412210848
To find the inclination of the hill to a horizontal plane, we can use trigonometry. Let's label the variables:
h = height of the tower (150 m)
d = distance from the point down the hill to the tower (650 m)
θ = angle between the surface of the hill and the line of sight to the top of the tower (12 degrees 30 minutes)
First, let's convert the angle from degrees and minutes to decimal degrees:
Angle in decimal degrees = 12 + (30/60) = 12.5 degrees
Now, we can calculate the vertical distance between the top of the tower and the point down the hill:
Vertical distance (v) = h - d * tan(θ)
Substituting the values:
v = 150 - 650 * tan(12.5)
Next, we can find the horizontal distance between the top of the tower and the point down the hill:
Horizontal distance (h) = d - v * tan(90 degrees - θ)
Substituting the values:
h = 650 - v * tan(90 - 12.5)
Finally, we can find the inclination of the hill to a horizontal plane:
Inclination = tan^(-1)(v / h)
Substituting the calculated values:
Inclination = tan^(-1)(150 - 650 * tan(12.5) / (650 - (150 - 650 * tan(12.5)) * tan(90 - 12.5)))
Using a calculator or mathematical software, you can calculate the exact value of the inclination.
To find the inclination of the hill to a horizontal plane, we can create a right triangle using the given information.
Let's label the various elements:
- Height of the tower: 150 meters
- Distance down the hill: 650 meters
- Angle between the surface of the hill and the line of sight to the top of the tower: 12 degrees 30 minutes
First, we need to find the length of the hypotenuse, which represents the line of sight from the point 650 meters down the hill to the top of the tower. We can use trigonometry to find the length of the hypotenuse.
Since we have an angle and one side of a right triangle, we can use the tangent function:
tangent(angle) = opposite / adjacent
Using this, we can solve for the opposite side:
tangent(12 degrees 30 minutes) = opposite / 650 meters
Now, let's calculate the value of the tangent:
tangent(12 degrees 30 minutes) = tan(12.5 degrees) = 0.224
Multiplying both sides of the equation by 650 meters, we get:
opposite = 650 meters * 0.224 = 145.6 meters
So, the opposite side (the difference in height) between the top of the tower and the point 650 meters down the hill is approximately 145.6 meters.
Next, we can calculate the difference in height between the starting point (at the bottom of the hill) and the point 650 meters down the hill. This will be equal to the height of the hill.
Height of the hill = height of the tower - opposite
Height of the hill = 150 meters - 145.6 meters = 4.4 meters
Therefore, the height of the hill is approximately 4.4 meters.
Finally, we can find the inclination of the hill to a horizontal plane by calculating the ratio of the height of the hill to the distance down the hill.
Inclination of the hill = height of the hill / distance down the hill
Inclination of the hill = 4.4 meters / 650 meters
Calculating this, we find:
Inclination of the hill = 0.006769
Finally, to express this inclination as an angle, we can use the inverse tangent function:
Angle = arctan(inclination of the hill)
Using a calculator, we find:
Angle = arctan(0.006769) = 0.388 degrees
Therefore, the inclination of the hill to a horizontal plane is approximately 0.388 degrees.