Please check these answers thanks.
1. The shadow of a vertical pole is 24.8 feet long when the angle of elevation of the sun is 42degrees 0.0'. Find the height of the pole.
Answer: 22.3=height
2. From a point on a lighthouse 108 feet above water, the angle of depression of a boat is 10degrees 10'. How far is the boat from the lighthouse?
Answer: 19.4=distance
3. What is the angle of elevation of the sun when a vertical pole 18 feet high casts a shadow of 11 feet long?
Answer: 36degrees=angle of elevation
4. In measuring the length of the east-west boundary of a piece of land, a surveyor comes to a marshy place. He diverts his path from a point P on the boundary line directly north to Q and from Q returns to R on the line. If the bearing of R from Q is S 60degrees 00'E and the distance PQ is 320 feet, find the distance PR.
Answer: 369.5= distance
2,3 wrong
I don't understand exactly what S60E means, it is not a standard measure. Is it 60 deg S of E, or 60 deg E of S?
1. To find the height of the pole, we can use the tangent function. The tangent of the angle of elevation is equal to the height of the pole divided by the length of the shadow. Let's calculate it:
tan(42 degrees 0.0') = height / 24.8 feet
Using a calculator, we have:
0.900404044297 = height / 24.8 feet
To solve for the height, we can cross multiply:
height = 0.900404044297 * 24.8 feet
Calculating this, we get:
height ≈ 22.35 feet
Therefore, the height of the pole is approximately 22.35 feet.
2. To find the distance of the boat from the lighthouse, we can use the tangent function again. The tangent of the angle of depression is equal to the height of the lighthouse divided by the distance of the boat from the lighthouse. Let's calculate it:
tan(10 degrees 10') = 108 feet / distance
Using a calculator, we have:
0.176326980708 = 108 feet / distance
To solve for the distance, we can cross multiply:
distance = 108 feet / 0.176326980708
Calculating this, we get:
distance ≈ 613.36 feet
Therefore, the boat is approximately 613.36 feet away from the lighthouse.
3. To find the angle of elevation of the sun, we can use the inverse tangent function. The inverse tangent of the height of the pole divided by the length of the shadow gives us the angle of elevation. Let's calculate it:
angle = arctan(18 feet / 11 feet)
Using a calculator, we have:
angle ≈ arctan(1.63636363636)
Calculating this, we get:
angle ≈ 35.97 degrees
Therefore, the angle of elevation of the sun is approximately 35.97 degrees.
4. To find the distance PR, we can use the law of cosines. Let's define the angle PQR as X.
Using the law of cosines, we have:
PR^2 = PQ^2 + QR^2 - 2 * PQ * QR * cos(X)
Plugging in the given values, we have:
PR^2 = 320 feet^2 + QR^2 - 2 * 320 feet * QR * cos(60 degrees)
Simplifying this equation, we have:
PR^2 = 102400 feet^2 + QR^2 - 640 * QR
Since PR is the distance and should be positive, we can take the positive square root:
PR = sqrt(102400 feet^2 + QR^2 - 640 * QR)
To find QR, we can use the law of sines. Let's define the angle QRP as Y.
Using the law of sines, we have:
QR / sin(Y) = PR / sin(X)
Plugging in the given values, we have:
QR / sin(60 degrees) = PR / sin(Y)
Simplifying this equation, we have:
QR = PR * sin(60 degrees) / sin(Y)
Now we can substitute QR in the previous equation:
PR = sqrt(102400 feet^2 + (PR * sin(60 degrees) / sin(Y))^2 - 640 * PR * sin(60 degrees) / sin(Y))
Solving this equation for PR requires numerical methods.
To verify these answers, let's go through each question and explain how to arrive at the correct answer:
1. The shadow of a vertical pole is 24.8 feet long when the angle of elevation of the sun is 42 degrees 0.0'. Find the height of the pole.
To find the height of the pole, we can use the trigonometric function tangent (tan). Tan is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the shadow length is the adjacent side, and the height of the pole is the opposite side.
Using the formula tan(angle) = opposite/adjacent, we can rearrange it to height = tan(angle) * adjacent.
So, the height of the pole is calculated as: height = tan(42° 0.0') * 24.8 = 22.3 feet.
Answer: The height of the pole is 22.3 feet.
2. From a point on a lighthouse 108 feet above water, the angle of depression of a boat is 10 degrees 10'. How far is the boat from the lighthouse?
To find the distance of the boat from the lighthouse, we can again use trigonometry. The tangent of the angle of depression is equal to the ratio of the opposite side (height of the lighthouse) to the adjacent side (distance to the boat).
Using the formula distance = opposite/tan(angle), we can calculate the distance as: distance = 108 / tan(10° 10') = 19.4 feet.
Answer: The boat is approximately 19.4 feet away from the lighthouse.
3. What is the angle of elevation of the sun when a vertical pole 18 feet high casts a shadow of 11 feet long?
To find the angle of elevation of the sun, we can use the inverse tangent function (arctan). Arctan calculates the angle that has a given ratio of the opposite and adjacent sides.
Therefore, the angle of elevation can be calculated as: angle = arctan(opposite/adjacent) = arctan(18/11) ≈ 36 degrees.
Answer: The angle of elevation of the sun is approximately 36 degrees.
4. In measuring the length of the east-west boundary of a piece of land, a surveyor diverts his path from point P on the boundary line directly north to Q and then returns to R on the line. If the bearing of R from Q is S 60 degrees 00' E and the distance PQ is 320 feet, find the distance PR.
To find the distance PR, we can use trigonometry and the concept of bearings. The bearing S 60 degrees 00' E can be interpreted as a clockwise angle of 60 degrees from the south direction.
Using the cosine rule, we can find the distance PR. In a triangle, the cosine of an angle is equal to the ratio of the adjacent side and the hypotenuse. In this case, PR is the hypotenuse, and PQ is the adjacent side.
Applying the formula PR^2 = PQ^2 + QR^2 - 2 * PQ * QR * cos(angle), we can substitute the known values and solve for PR.
PR^2 = 320^2 + QR^2 - 2 * 320 * QR * cos(60°) = 102400 + QR^2 - 320 * QR
Since we want to find PR, we can set the equation equal to zero and solve it. QR represents the distance between Q and R, which is the value we are looking for.
In this case, we will need additional information about the distance QR or further calculations to solve for PR.
Hope this helps in understanding the process for solving these types of problems!