shadow of a goal post is 6m longer when the elevation of the sun is 30° then when it is 60° calculate the hight of the post?
5.2m
To solve this problem, let's assume the height of the goal post is represented by 'h' meters.
When the elevation of the sun is 30°, the length of the shadow cast by the goal post can be represented as 'x' meters. We can form the following equation:
tan(30°) = h / x
Simplifying this equation, we have:
√3/3 = h / x
Multiplying both sides by 'x', we get:
√3x = 3h
Now, let's consider when the elevation of the sun is 60°. The length of the shadow cast by the goal post can be represented as 'x + 6' meters. We can form the following equation:
tan(60°) = h / (x + 6)
Simplifying this equation, we have:
√3 = h / (x + 6)
Multiplying both sides by '(x + 6)', we get:
√3(x + 6) = h
Now we have a system of two equations:
1) √3x = 3h
2) √3(x + 6) = h
We can solve this system of equations by substitution or elimination method.
Let's solve it using substitution method:
From equation 1, we can express h in terms of x:
h = √3x / 3
Substituting this value of h in equation 2, we get:
√3(x + 6) = √3x / 3
Multiplying both sides by 3 to eliminate the denominator:
3√3(x + 6) = √3x
Expanding both sides:
3√3x + 18√3 = √3x
Subtracting √3x from both sides:
3√3x - √3x + 18√3 = 0
Simplifying:
2√3x = -18√3
Dividing both sides by 2√3:
x = -18
Since the length of a shadow cannot be negative, there seems to be an error in the problem statement. Please check for any mistakes or missing information.
If you have any further questions, please, let me know.
To calculate the height of the goal post, we can use trigonometry and set up a proportion based on the given information.
Let's assume the height of the goal post is represented by "h" meters.
When the elevation of the sun is 30°, the shadow of the goal post is 6m longer than its height. This means that the shadow length would be h + 6.
When the elevation of the sun is 60°, we need to calculate the new shadow length.
We can use the trigonometric tangent function (tan) to relate the angle of elevation to the shadow length. The tangent of an angle is equal to the opposite side divided by the adjacent side of a right triangle.
In this case, the height of the post is the opposite side, and the shadow length is the adjacent side.
When the sun's elevation is 30°, the tangent of the angle is given by:
tan(30°) = h / (h + 6)
We can solve this equation to find the value of h:
tan(30°) = h / (h + 6)
√(3) / 3 = h / (h + 6)
√(3)(h + 6) = 3h (cross-multiplication)
√(3)h + 6√(3) = 3h
6√(3) = 3h - √(3)h
6√(3) = (3 - √(3))h
h = (6√(3)) / (3 - √(3))
Now that we have found the value of h, we can substitute it to find the height of the goal post.
h ≈ 5.858 meters (rounded to three decimal places)
Therefore, the height of the goal post is approximately 5.858 meters.