From a point A, level with the foot of a hill, the angle of elevation of the top of the hill is 16. From a point B, 950 m clearer the foot of the hill, the angle of elevation of the top is 35. Determine the height of the hill.
Tan35 = h/d, h = d*Tan35.
Tan16 = h/(d+950), h = (d+950)Tan16.
d*Tan35 = (d+950)*Tan16.
0.7d = 0.287d + 272.4, 0.413d = 272.4, d = 659.6 m.
h = d*Tan35 = 659.6 * Tan35 = 461.8 m.
To determine the height of the hill, we can use the trigonometric concept of tangent.
Let's denote the height of the hill as 'h' (in meters). We are given two angles of elevations: 16 degrees from point A and 35 degrees from point B.
From point A, we can consider the right-angled triangle ABC, where ABC represents the hill, AB represents the base, and BC represents the height.
Using trigonometric ratios, we can write the following equation:
tan(16) = BC / AB
Similarly, from point B, we can consider the right-angled triangle BCD, where BCD represents the extended portion of the hill beyond point B, BD represents the base, and CD represents the height.
Using trigonometric ratios, we can write the following equation:
tan(35) = CD / BD
Since we have to find the height of the hill, we need to find the value of BC + CD.
BC + CD = AB + BD
Now, let's solve these equations step by step:
1. Calculate AB:
To find AB, we can use the trigonometric ratio for the angle of elevation of 16 degrees:
tan(16) = BC / AB
Rearranging the equation:
AB = BC / tan(16)
2. Calculate BD:
To find BD, we can use the trigonometric ratio for the angle of elevation of 35 degrees:
tan(35) = CD / BD
Rearranging the equation:
BD = CD / tan(35)
3. Calculate the sum of AB and BD:
Substituting the values of AB and BD into the equation BC + CD = AB + BD:
BC + CD = BC / tan(16) + CD / tan(35)
4. Simplify the equation:
To simplify the equation, we combine like terms:
BC + CD = BC * (tan(35) / tan(16)) + CD / tan(35)
5. Solve for BC + CD (AB + BD):
Now, we can solve for BC + CD by subtracting BC from both sides of the equation:
CD = BC * (tan(35) / tan(16)) - BC
Simplifying further:
CD = BC * (tan(35) / tan(16) - 1)
Now, we have an equation to calculate CD, which represents the height of the hill.
6. Calculate CD (height of the hill):
We know that BC + CD represents the total height of the hill. Given that BC is the distance between points A and B (950 meters), we can substitute the values into the equation:
CD = 950 * (tan(35) / tan(16) - 1)
By evaluating this equation, we can find the exact height of the hill.