1) A student can see a tower from the closet point of the soccer field at his high school. The edge of the soccer field is about 105 feet from the water tower and the water tower stands at a height of 51 feet. What is the angle of elevation from the student's feet?
2) A roofer props a ladder against a wall so that the base of the ladder is 4 feet away from the building. If the angle of elevation from the bottom of the ladder to the roof is 63", how long is the ladder?
3) A meteorologist measures the angle of depression of a weather balloon as 53°. A radio signal from the balloon indicates that it is 1680 feet from his receiver. How high is the weather balloon from the ground?
4) A blimp is flying to cover a football game. The pilot sights the stadium at an 8" angle of depression. The blimp is flying at an altitude of 420m. How many meters is the blimp from the stadium?
5) A sledding run is 280 yards long with a vertical drop of 18.5 yards. What is the angle of depression of the run?
6) A golfer is standing at the tee, looking up to the green on a hill. If the tee is 38 yards lower than the green and the angle of elevation is 15", find the distance from the tee to the hole?
Thank you
As always, draw a diagram, and review your basic trig functions.
(1) tanθ = 51/150
(2) 4/x = cos63°
(3) x/1680 = sin53°
(4) x/420 = tan8"
(5) sinθ = 18.5/280
(6) 38/x = sin15°
1) To find the angle of elevation from the student's feet, we can use trigonometry. The tangent function can be used in this case.
First, let's label the given information:
- Distance from the edge of the soccer field to the water tower (adjacent side): 105 feet
- Height of the water tower (opposite side): 51 feet
We can calculate the angle of elevation using the formula: tan(angle) = opposite/adjacent.
In this case, tan(angle) = 51/105.
To find the angle, we can take the inverse tangent of both sides: angle = arctan(51/105).
Using a calculator, we can find the value of the angle.
2) To find the length of the ladder, we can again use trigonometry. This time, we will use the sine function.
Given information:
- Distance from the base of the ladder to the building (adjacent side): 4 feet
- Angle of elevation from the bottom of the ladder to the roof (angle): 63 degrees (note the typo in your question, it should be degrees, not inches)
We can use the sine function: sin(angle) = opposite/hypotenuse.
In this case, sin(63) = opposite/4.
To find the length of the ladder, we can rearrange the equation: opposite = sin(63) * 4.
Using a calculator, we can find the value of the opposite side, which represents the length of the ladder.
3) In this problem, we have the angle of depression and the distance from the receiver to the balloon. We need to find the height of the weather balloon.
Given information:
- Angle of depression of the weather balloon (angle): 53 degrees
- Distance from the receiver to the balloon (opposite side): 1680 feet
We can use the tangent function to find the height of the weather balloon.
tan(angle) = opposite/adjacent.
In this case, tan(53) = height/1680.
To find the height, we multiply both sides by 1680: height = tan(53) * 1680.
Using a calculator, we can calculate the value of the height.
4) Similar to the previous problem, we have the angle of depression and the altitude of the blimp. We need to find the distance of the blimp from the stadium.
Given information:
- Angle of depression of the stadium (angle): 8 degrees
- Altitude of the blimp (opposite side): 420 meters
Again, we use the tangent function to find the distance from the blimp to the stadium.
tan(angle) = opposite/adjacent.
In this case, tan(8) = opposite/adjacent.
To find the distance, we rearrange the equation: adjacent = opposite/tan(8).
Using a calculator, we can calculate the value of the adjacent side, which represents the distance from the blimp to the stadium.
5) In this problem, we have the length of the sledding run and the vertical drop. We need to find the angle of depression.
Given information:
- Length of the sledding run (adjacent side): 280 yards
- Vertical drop (opposite side): 18.5 yards
We can use the inverse tangent function to find the angle of depression.
angle = arctan(opposite/adjacent).
In this case, angle = arctan(18.5/280).
Using a calculator, we can find the value of the angle.
6) Finally, we have the golfer's position, the height of the green, and the angle of elevation. We need to find the distance from the tee to the hole.
Given information:
- Height of the green (opposite side): unknown
- Difference in elevation from the tee to the green (adjacent side): 38 yards
- Angle of elevation (angle): 15 degrees
We can use the tangent function again to find the distance.
tan(angle) = opposite/adjacent.
In this case, tan(15) = opposite/38.
To find the distance, we can rearrange the equation: opposite = tan(15) * 38.
Using a calculator, we can calculate the value of the opposite side, which represents the distance from the tee to the hole.
1) To find the angle of elevation from the student's feet, we can use trigonometry. In this case, we have the opposite side (height of the water tower) and the adjacent side (distance from the student to the tower).
We can use the tangent function, which is defined as the opposite side divided by the adjacent side.
So, tan(angle) = opposite/adjacent = 51/105
Now we can calculate the angle of elevation:
angle = arctan(51/105) ≈ 25.39°
Therefore, the angle of elevation from the student's feet is approximately 25.39°.
2) To find the length of the ladder, we can use trigonometry. In this case, we have the opposite side (height of the building) and the adjacent side (distance from the base of the ladder to the building).
We can use the tangent function again:
tan(angle) = opposite/adjacent
tan(63°) = height/4
Now we can solve for the height:
height = 4 * tan(63°) ≈ 7.67 feet
Therefore, the length of the ladder is approximately 7.67 feet.
3) To find the height of the weather balloon, we can use trigonometry. In this case, we have the opposite side (height from the ground to the balloon) and the adjacent side (distance from the receiver to the balloon).
We can use the tangent function once again:
tan(angle) = opposite/adjacent
tan(53°) = opposite/1680
Now we can solve for the height:
opposite = 1680 * tan(53°) ≈ 2068.8 feet
Therefore, the height of the weather balloon from the ground is approximately 2068.8 feet.
4) To find the distance from the blimp to the stadium, we can use trigonometry. In this case, we have the opposite side (height of the blimp) and the angle of depression.
We can use the tangent function:
tan(angle) = opposite/adjacent
tan(8°) = 420/adjacent
Now we can solve for the adjacent side (distance from the blimp to the stadium):
adjacent = 420 / tan(8°) ≈ 2930.71 meters
Therefore, the blimp is approximately 2930.71 meters from the stadium.
5) To find the angle of depression of the sledding run, we can use trigonometry. In this case, we have the opposite side (vertical drop) and the adjacent side (length of the run).
We can use the inverse tangent function:
angle = arctan(opposite/adjacent)
angle = arctan(18.5/280) ≈ 3.79°
Therefore, the angle of depression of the run is approximately 3.79°.
6) To find the distance from the tee to the hole, we can use trigonometry. In this case, we have the opposite side (height difference between the tee and the green) and the angle of elevation.
We can use the tangent function:
tan(angle) = opposite/adjacent
tan(15°) = 38/x
Now we can solve for the adjacent side (distance from the tee to the hole):
x = 38 / tan(15°) ≈ 138.15 yards
Therefore, the distance from the tee to the hole is approximately 138.15 yards.