When the sun's angle of elevation is 30 degree the shadow of a post is 6 ft longer than when the angle is 45

degree. find the height of the post

let the height of the post be h ft, let the length of the shadow be x

when angle is 45°, the height of the post and its shadow would be equal, so x = h at 45°

at 30°, length of shadow = h+6

tan 30 = h/(h+6)
1/√3 = h/(h+6)
√3h = h + 6
√3h - h = 6
h(√3 - 1) = 6
h = 6/(√3 - 1) = appr. 8.2 ft
√3h - h =

To find the height of the post, we can use trigonometry.

Let's assume the height of the post is "h" feet.

When the sun's angle of elevation is 30 degrees, we can form a right triangle with the post, the shadow, and the sun's rays. In this triangle, the opposite side is the height of the post (h), and the adjacent side is the length of the shadow.

Using the trigonometric function tangent (tan), we can write the equation:

tan(30) = h / (x + 6)

Where x is the length of the shadow when the angle of elevation is 30 degrees.

When the sun's angle of elevation is 45 degrees, we can also form a right triangle with the post, the shadow, and the sun's rays. In this triangle, the opposite side is the height of the post (h), and the adjacent side is the length of the shadow.

Using the same trigonometric function tangent (tan), we can write the equation:

tan(45) = h / x

Where x is the length of the shadow when the angle of elevation is 45 degrees.

Now, we can set up a system of equations using the above equations:

tan(30) = h / (x + 6)
tan(45) = h / x

We can simplify the equations:

sqrt(3) = h / (x + 6)
1 = h / x

Now, we can solve the system of equations. We can rearrange the second equation to isolate h:

h = x

Substituting this value of h into the first equation, we get:

sqrt(3) = x / (x + 6)

To solve for x, we can square both sides of the equation:

3 = x^2 / (x + 6)^2

Cross multiplying, we get:

3(x + 6)^2 = x^2

Expanding and rearranging the equation:

3(x^2 + 12x + 36) = x^2
3x^2 + 36x + 108 = x^2
2x^2 - 36x - 108 = 0

Simplifying, we get a quadratic equation:

x^2 - 18x - 54 = 0

Now, we can solve this equation using factoring, completing the square, or using the quadratic formula. Solving it, we find the value of x:

x = 20.61 ft (approx)

Since the question asks for the height of the post, we can simply substitute this value of x back into one of the earlier equations:

h = x = 20.61 ft

Therefore, the height of the post is approximately 20.61 feet.