Can you please check these answers thanks
1. A ladder 14 feet long is leaning against a house. The foot of the ladder is 4.8 feet from the house. Find the angle of elevation of the ladder and the height it reaches on the house.
Answer: 13.2=height
18.9 =angle of elevation
2. From a point 230 feet from the base of a smokestack, the angle of elevation of the top is 28degrees 30'. Find the height of the smokestack.
Answer: 124.9= height
3. From a tower 70 feet high, the angle of depression of a car is 32degrees. How far is the car from the tower?
Answer: 132.1=distance
4. From a cliff 130 feet above the shore line, the angle of depression of a ship is 22degrees 40'. Find the distance from the ship to a point on the shore directly below the observer.
Answer: 54.3=distance
1. To find the angle of elevation of the ladder, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height the ladder reaches on the house, which we need to find, and the adjacent side is the distance between the foot of the ladder and the house, given as 4.8 feet. So, we have:
tan(angle of elevation) = height / 4.8
To find the angle of elevation, we need to take the arctangent (inverse tangent) of both sides:
angle of elevation = arctan(height / 4.8)
To find the height the ladder reaches on the house, we can use the Pythagorean theorem. The hypotenuse of the right triangle formed by the ladder, the height on the house, and the distance from the foot of the ladder to the house is given as 14 feet. So, we have:
height^2 + 4.8^2 = 14^2
Now we can calculate the height and angle of elevation using these equations.
2. To find the height of the smokestack, we can use the tangent function again. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the smokestack, which we need to find, and the adjacent side is the distance from the point to the base of the smokestack, given as 230 feet. So, we have:
tan(28 degrees 30') = height / 230
To find the height, we can multiply both sides by 230 and then take the tangent of 28 degrees 30'.
3. The angle of depression of the car is given as 32 degrees. To find the distance from the car to the tower, we can use the tangent function once again. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the tower, given as 70 feet, and the adjacent side is the distance from the car to the tower, which we need to find. So, we have:
tan(32 degrees) = 70 / distance
To find the distance, we can multiply both sides by the distance and then take the tangent of 32 degrees.
4. The angle of depression of the ship is given as 22 degrees 40'. To find the distance from the ship to a point on the shore directly below the observer, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the cliff, given as 130 feet, and the adjacent side is the distance from the observer to the point on the shore, which we need to find. So, we have:
tan(22 degrees 40') = 130 / distance
To find the distance, we can multiply both sides by the distance and then take the tangent of 22 degrees 40'.
To verify the given answers, we can use trigonometric ratios and calculations. Let's go through each question and explain the steps to find the answers.
1. A ladder 14 feet long is leaning against a house. The foot of the ladder is 4.8 feet from the house. Find the angle of elevation of the ladder and the height it reaches on the house.
To find the height reached on the house, we can use the trigonometric function sine. The opposite side of the angle of elevation is the height, and the hypotenuse is the ladder itself.
Using the formula:
sin(angle) = opposite / hypotenuse
We can solve for the angle of elevation:
sin(angle) = height / ladder length
sin(angle) = 13.2 / 14
angle = arcsin(13.2 / 14)
angle ≈ 54.3 degrees
Hence, the angle of elevation is approximately 54.3 degrees.
For the height reached on the house, we can use the trigonometric function cosine. The adjacent side of the angle of elevation is the distance from the foot of the ladder to the house.
Using the formula:
cos(angle) = adjacent / hypotenuse
We can solve for the height:
cos(angle) = distance to the house / ladder length
cos(angle) = 4.8 / 14
height = cos(angle) * ladder length
height ≈ 4.8 * cos(54.3)
height ≈ 13.2 feet
Therefore, the ladder reaches a height of approximately 13.2 feet on the house.
2. From a point 230 feet from the base of a smokestack, the angle of elevation of the top is 28 degrees 30'. Find the height of the smokestack.
Using the tangent function:
tan(angle) = opposite / adjacent
We can solve for the height:
tan(angle) = height / distance to the smokestack
tan(angle) = height / 230
height = tan(angle) * 230
height ≈ tan(28.5) * 230
height ≈ 124.9 feet
Therefore, the height of the smokestack is approximately 124.9 feet.
3. From a tower 70 feet high, the angle of depression of a car is 32 degrees. How far is the car from the tower?
To find the distance from the car to the tower, we can use the tangent function:
tan(angle) = opposite / adjacent
We can solve for the distance:
tan(angle) = opposite / height of the tower
tan(angle) = distance / 70
distance = tan(angle) * 70
distance ≈ tan(32) * 70
distance ≈ 132.1 feet
So, the car is approximately 132.1 feet away from the tower.
4. From a cliff 130 feet above the shoreline, the angle of depression of a ship is 22 degrees 40'. Find the distance from the ship to a point on the shore directly below the observer.
Using the tangent function:
tan(angle) = opposite / adjacent
We can solve for the distance:
tan(angle) = height of the cliff / distance to the shore point
tan(angle) = 130 / distance
distance = 130 / tan(angle)
distance ≈ 130 / tan(22.67)
distance ≈ 54.3 feet
Therefore, the distance from the ship to a point on the shore directly below the observer is approximately 54.3 feet.
These calculations support the given answers for each of the four problems.