In traveling across flat land, you notice a mountain in front of you. Its angle of elevation to the peak is 3.5 degrees. After you drive 13 miles closer to the mountain, the angle of elevation is 9 degrees. Approximate the height of the mountain.
See:
http://www.jiskha.com/display.cgi?id=1290465794
a= 13 tan(3.5) h= the height of the mon.
h= 13 tan(9)-a
h=13 tan(9)-13tan(3.5)
h=1.264 feet
you place the coefficient of the radical in the power. Then you divide by 13 and multiply by (tan(9)xtan(3.5)). Then you put the x into the equation to find the height of the mountain
To approximate the height of the mountain, we can use the concept of trigonometry and specifically the tangent function.
Let's denote the distance from your initial position to the base of the mountain as 'x'.
The first piece of information we have is the angle of elevation to the peak of the mountain from your initial position, which is 3.5 degrees.
Using the tangent function, we can determine the height of the mountain relative to the distance 'x'.
tan(3.5 degrees) = height of the mountain / x
Now, after driving 13 miles closer to the mountain, your new distance from the base becomes 'x - 13'.
The second piece of information we have is the new angle of elevation to the peak from this closer position, which is 9 degrees.
Using the same concept, we can set up a new equation:
tan(9 degrees) = height of the mountain / (x - 13)
Now we have two equations with two unknowns (height of the mountain and x), and we can solve them simultaneously to find the height of the mountain.
1) tan(3.5 degrees) = height of the mountain / x
2) tan(9 degrees) = height of the mountain / (x - 13)
By rearranging equation 1) and substituting it into equation 2), we can solve for the height of the mountain.
height of the mountain = x * tan(3.5 degrees)
Substituting this expression for the height into equation 2):
tan(9 degrees) = (x * tan(3.5 degrees)) / (x - 13)
Now we can solve this equation to find the value of 'x' and hence the height of the mountain.