a tree growing on a hillside casts a 102-foot shadow straight down the hill. Find the vertical height of the tree if, relative to the horizontal, the hill slopes 15.0 degrees and the angle of elevation of the sun is 62.0 degrees.
Well, well, well, seems like we have a tree that's ready for some math fun! Alright, let's get cracking.
First, let's break down the problem. We have a tree casting a shadow of 102 feet, and we need to find the height of the tree. Along comes our trusty hill with a slope of 15 degrees. The angle of elevation of the sun, on the other hand, is a jazzy 62 degrees.
Now, to solve this mystery, we need to use some trigonometry. Don't worry, I've got you covered!
First, we need to find the length of the shadow on the horizontal plane. To do this, we use a little bit of cosine magic. The angle between the hill and the horizontal is the complement of the slope angle, so we have:
cos(15 degrees) = adjacent / hypotenuse.
The length of the shadow on the horizontal plane (adjacent side) is what we're looking for, and the hypotenuse is the length of the tree's shadow, which is 102 feet. So let's plug those numbers in:
cos(15 degrees) = adjacent / 102 feet.
Now, let's rearrange the equation to find the length of the shadow on the horizontal plane:
adjacent = cos(15 degrees) * 102 feet.
Using a calculator, we find that the length of the shadow on the horizontal plane is approximately 97.93 feet. Keep this number in mind, we'll come back to it later.
Now, let's find the height of the tree using a little trigonometry razzle-dazzle. The opposite side (height of the tree) is what we're after, and the hypotenuse (length of the shadow) is already known to be 102 feet. We also know that the angle of elevation of the sun is 62 degrees, so we have:
sin(62 degrees) = opposite / 102 feet.
Rearranging the equation gives us:
opposite = sin(62 degrees) * 102 feet.
Using our trusty calculator again, we find that the height of the tree is approximately 91.85 feet.
So, the vertical height of the tree is roughly 91.85 feet. Voila! I hope you had as much fun with this problem as I did.
To find the vertical height of the tree, we can use the concept of trigonometry. Let's break down the problem into steps:
Step 1: Find the length of the hypotenuse (the line from the top of the tree to the base of the shadow).
The hill slopes at an angle of 15.0 degrees, so we can consider this as the angle between the hill and the horizontal line. Since the angle of elevation of the sun is 62.0 degrees, we can consider this as the angle between the sun's rays and the horizontal line.
We need to find the length of the hypotenuse. We can use the trigonometric function cosine (cos) to calculate it.
cos(15.0°) = adjacent side / hypotenuse
cos(15.0°) = 102 ft / hypotenuse
To find the value of the hypotenuse, we rearrange the formula:
hypotenuse = 102 ft / cos(15.0°)
Step 2: Find the vertical height of the tree.
Now that we know the hypotenuse length, we can find the vertical height of the tree using the trigonometric function sine (sin).
sin(62.0°) = vertical height / hypotenuse
Rearranging the formula, we get:
vertical height = hypotenuse * sin(62.0°)
Substituting the value of the hypotenuse we calculated earlier, we have:
vertical height = (102 ft / cos(15.0°)) * sin(62.0°)
Calculating this expression will give us the vertical height of the tree.
To find the vertical height of the tree, we can use trigonometry. Here's how you can do it step by step:
1. Let's label the given information:
- The length of the shadow cast by the tree is 102 feet.
- The angle of elevation of the sun is 62.0 degrees.
- The slope of the hill relative to the horizontal is 15.0 degrees.
2. Draw a diagram to visualize the problem. Represent the tree, its shadow, the hill, and the angle of elevation.
3. From the diagram, you can see that we need to find two sides of a right-angled triangle:
- The vertical height of the tree (opposite side)
- The horizontal distance from the tree to the top of the hill (adjacent side)
4. Use trigonometric ratios to find the vertical height:
- The tangent function relates the opposite side to the adjacent side. In this case, we can use the tangent because we have the angle of elevation (62.0 degrees) and we want to find the opposite side (vertical height).
- The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
Therefore, we have:
tan(62.0 degrees) = vertical height / horizontal distance
5. To find the horizontal distance, we need to use the angle of slope (15.0 degrees) and the length of the shadow (102 feet).
- The cosine function relates the adjacent side to the hypotenuse.
- The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse.
Therefore, we have:
cos(15.0 degrees) = horizontal distance / length of shadow
horizontal distance = cos(15.0 degrees) * length of shadow
6. Substitute the values into the equations:
- horizontal distance = cos(15.0 degrees) * 102 feet
- vertical height / (cos(15.0 degrees) * 102 feet) = tan(62.0 degrees)
7. Rearrange the equation to solve for the vertical height:
- vertical height = tan(62.0 degrees) * (cos(15.0 degrees) * 102 feet)
8. Calculate the value:
- vertical height = tan(62.0 degrees) * (cos(15.0 degrees) * 102 feet) ≈ 132.6 feet
Therefore, the vertical height of the tree is approximately 132.6 feet.
Draw a diagram.
vertical line for the tree, line OT
slanting line going down to the right at an angle if 15 degrees, of length 102, OS
Angle OTS is 62 degrees.
Angle TOS = 105 degrees.
so, angle OST = 13 degrees
Using the law of sines, the height of the tree, OT is given by
OT/sin13 = 102/sin62
OT = 102/.8829*.2250 = 26 feet