A tree growing on a slope of 13 degree to the horizontal is 14m tall. Find the length of the shadow it casts up the hill when the sun is at an angle of elevation of 30 degree. (The shadow is cast directly up the slope by the sun)
Draw a suitable diagram, and you can see that the shadow length x is given by
x/sin60 = 14/sin43
To solve this problem, we will use trigonometry and the concept of similar triangles.
Let's draw a diagram to visualize the situation:
/|
h / |14m
/ |
/ |
/____|
d
In the diagram, h represents the height of the tree, and d represents the length of the shadow cast up the hill. We are given that the slope of the hill is 13 degrees, and the angle of elevation of the sun is 30 degrees.
To find the length of the shadow (d), we can use the tangent function:
tan(30) = h/d
Now, we need to find the value of h. We can use the given information about the slope of the tree.
We know that the slope of the tree is equal to the height of the tree divided by the distance along the ground. In this case, the slope is the opposite side divided by the adjacent side of a right triangle. Therefore, we can use the tangent function:
tan(13) = h/14m
To find h, we can rearrange the equation:
h = 14m * tan(13)
Now we can substitute this value into our original equation to find the length of the shadow (d):
tan(30) = (14m * tan(13)) / d
To solve for d, we can rearrange the equation:
d = (14m * tan(13)) / tan(30)
Now we can calculate d using a calculator:
d = (14 * tan(13)) / tan(30)
Plugging in the values and evaluating the expression, we get:
d ≈ 7.99m
Therefore, the length of the shadow cast up the hill by the tree is approximately 7.99 meters.