A tree grows vertically on the side of a hill and casts a shadow of 12.2m down the hill when the angle of the sun is 35 degrees. The angle between the tree and it's shadow is 101 degrees.
Determine the height of the tree, to the nearest hundredth of a metre.
Draw a diagram. Use the law of sines to find the height h, since
h/sin24° = 12.2/sin55°
To determine the height of the tree, we can use the properties of trigonometry and create a right triangle with the tree, its shadow, and a line perpendicular to the ground.
Let's break down the given information:
- The shadow of the tree is 12.2m.
- The angle between the tree and its shadow is 101 degrees.
- The angle of the sun is 35 degrees.
To solve this problem, we can start by finding the length of the adjacent side of the right triangle using the angle of the sun and the length of the shadow. The adjacent side represents the distance from the base of the tree to the tip of its shadow.
Using trigonometry, we can apply the cosine function since we have the adjacent side and the hypotenuse of the right triangle.
cos(angle) = adjacent / hypotenuse
cos(35 degrees) = adjacent / 12.2m
To solve for the adjacent side, we rearrange the equation:
adjacent = cos(35 degrees) * 12.2m
Now we can calculate the value of the adjacent side.
adjacent = cos(35 degrees) * 12.2m = 12.2 * cos(35 degrees) = 9.997m (rounded to three decimal places)
The adjacent side represents the height from the base of the tree to the tip of its shadow.
Next, we need to find the opposite side of the triangle, which represents the height of the tree.
Again, using trigonometry, we can apply the sine function since we have the opposite side and the hypotenuse of the right triangle.
sin(angle) = opposite / hypotenuse
sin(101 degrees) = opposite / 12.2m
To solve for the opposite side, we rearrange the equation:
opposite = sin(101 degrees) * 12.2m
Now we can calculate the value of the opposite side.
opposite = sin(101 degrees) * 12.2m = 12.2 * sin(101 degrees) = 12.076m (rounded to three decimal places)
The opposite side represents the height of the tree.
Therefore, the height of the tree is approximately 12.076m to the nearest hundredth.