Question 1

Solve the problem.
-At an altitude of 3300 ft, the engine on a small plane fails. What angle of glide is needed to reach an airport runway that is 4 miles away by land? (Round your answer to the nearest tenth of a degree.)

Accepted Answer

To find the glide angle, we can use the tangent function in trigonometry, where the tangent of the angle is equal to the opposite side (altitude) over the adjacent side (distance to the airport). First, convert the distance to feet since the altitude is given in feet (1 mile = 5280 feet). So, 4 miles = 4 * 5280 = 21120 feet. Then, $\tan(\theta) = \frac{3300}{21120}$. Solving for $\theta$ gives us an angle of approximately 8.9 degrees.

Question 2

Solve the problem.
-From a distance of 48 feet from the base of a building, the angle of elevation to the top of the building is 69°. Estimate the height of the building to the nearest foot.

Accepted Answer

Using the tangent function, tan(69°) = height / 48. Solving for height gives height = 48 * tan(69°) ≈ 125 ft.

Question 3

Solve the problem.
-A blimp is 1700 meters high in the air and measures the angles of depression to two stadiums to the west of the blimp. If those measurements are 72.4° and 27.8°, how far apart are the two stadiums to the nearest meter?

Accepted Answer

To find the distance between the two stadiums, we use the tangent of the angles of depression. For the first stadium, tan(72.4°) = 1700/d1, and for the second stadium, tan(27.8°) = 1700/d2. Solving these gives d1 and d2, and the distance between the stadiums is |d2 - d1|, which is approximately 2685 meters.

Question 4

Solve the problem.
-Your math class is going to test new digital clinometers by measuring the angle of elevation of a kite you will fly. The kite flies to an angle of 68.9° on 360 feet of string. Assuming that the string is taut, how high is the kite
To the nearest foot?

Accepted Answer

The height of the kite can be found using trigonometry, specifically the sine function, since we have the angle of elevation and the length of the string. The sine of an angle in a right triangle is equal to the opposite side (height of the kite) divided by the hypotenuse (length of the string). So, sin(68.9°) = height / 360. Solving for height gives height = 360 * sin(68.9°). Using a calculator, this gives a height of approximately 336 feet.

Question 5

Solve the problem.
-The air speed of an airplane is 670 km/hr and its angle of climb is 6.08°. What is its ground speed (to the nearest km/hr)?

Accepted Answer

The ground speed of the airplane can be found by multiplying its air speed by the cosine of its angle of climb. Using the given air speed of 670 km/hr and an angle of climb of 6.08°, the calculation is as follows: ground speed = 670 * cos(6.08°). The cosine of 6.08 degrees is approximately 0.994, so the ground speed is approximately 666 km/hr.

Question 6

Solve the problem.
-When sitting atop a tree and looking down at his pal Joey, the angle of depression of Mack's line of sight is 35° 22'. If Joey is known to be standing 31 feet from the base of the tree, how tall is the tree (to the nearest foot)?

Accepted Answer

The answer of Solve the problem.
-When sitting atop a tree...

Question 7

Solve the problem.
-In one area, the lowest angle of elevation of the sun in winter is 22° 19'. A fence is to be built 13 ft away from a plant in the direction of the sun. (See drawing) Find the maximum height, x , for the fence so that the plant will
Get full sun. Round your answer to the tenths place when necessary.

Accepted Answer

The answer of Solve the problem.
-In one area, the lowest...

Question 8

Solve the problem.
-A kite is currently flying at an altitude of 15 meters above the ground. If the angle of elevation from the ground to the kite is 30°, find the length of the kite string to the nearest meter.

Accepted Answer

The answer of Solve the problem.
-A kite is currently flying...

Question 9

Solve the problem.
-A police helicopter is monitoring the speed of two cars on a straight road. The helicopter is at an altitude of 4500 feet directly above the road. At one instant, the angle of elevation from the first car to the helicopter is 20°, and
The angle of elevation from the second car to the helicopter is 15°. How far apart are the two cars to the nearest
Foot?

Accepted Answer

To find the distance between the two cars, we can use the tangent of the angles of elevation and the altitude of the helicopter. For the first car, the distance from the point directly below the helicopter (let's call this point H) to the first car (C1) can be found using the tangent of 20°: $\tan(20°) = \frac{opposite}{adjacent} = \frac{4500}{HC1}$. Solving for $HC1$, we get $HC1 = \frac{4500}{\tan(20°)}$.For the second car (C2), using the tangent of 15°: $\tan(15°) = \frac{4500}{HC2}$. Solving for $HC2$, we get $HC2 = \frac{4500}{\tan(15°)}$.The distance between the two cars is the difference between $HC2$ and $HC1$: $Distance = HC2 - HC1$.Plugging in the values and calculating, we find that the distance is approximately 4431 feet, rounding to the nearest foot.

Question 10

Solve the problem.
-From the top of a vertical tower, 353 feet above the the surface of the earth, the angle of depression to a dog house is 28° 51'. How far is it from the dog house to the foot of the tower? Round your answer to the hundredths
Place.

Accepted Answer

The answer of Solve the problem.
-From the top of a...

Question 11

Solve the problem.
-A vertical antenna is mounted atop a 23-ft pole. From a point on the level ground 57 feet from the base of the pole, the antenna subtends an angle of 10.1°. Find the length of the antenna to the nearest tenth of a foot.

Accepted Answer

The length of the antenna can be found using the tangent of the angle it subtends from the point on the ground. The tangent of an angle in a right triangle is the ratio of the opposite side (the length of the antenna plus the pole) to the adjacent side (the distance from the point to the base of the pole). Thus, $\tan(10.1°) = \frac{\text{antenna length} + 23}{57}$. Solving for the antenna length gives approximately 12.7 ft.

Question 12

Solve the problem.
-The index of refraction for air, Ia, is 1.0003. The index of refraction for water,  = 31.5°, find W to the nearest tenth.

Accepted Answer

The answer of Solve the problem.
-The index of refraction for...

Question 13

Solve the problem.
-A 38-foot ladder is leaning against the side of a building. If the ladder makes an angle of 23° 30' with the side of the building, how far up from the ground does the ladder make contact with the building? Round your answer
To the hundredths place.

Accepted Answer

The answer of Solve the problem.
-A 38-foot ladder is leaning...

Question 14

Solve the problem.
-From a balloon 1197 feet high, the angle of depression to the ranger headquarters is 50°21'. How far is the headquarters from a point on the ground directly below the balloon (to the nearest foot)?

Accepted Answer

To find the distance from the point directly below the balloon to the ranger headquarters, we can use the tangent of the angle of depression, which is equal to the opposite side (the height of the balloon) over the adjacent side (the distance we're trying to find). The tangent of 50°21' can be calculated or looked up, and the formula to use is:$$ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} $$Given:- $\theta = 50°21'$- Opposite (height of the balloon) = 1197 ftRearranging the formula to solve for the adjacent side (distance to headquarters), we get:$$ \text{Adjacent} = \frac{\text{opposite}}{\tan(\theta)} $$Plugging in the values:$$ \text{Adjacent} = \frac{1197}{\tan(50°21')} $$Using a calculator to find $\tan(50°21')$ and solving gives a distance of approximately 992 feet to the nearest foot.

Question 15

Solve the problem.
-A contractor needs to know the height of a building to estimate the cost of a job. From a point 86 feet away from the base of the building, the angle of elevation to the top of the building is found to be 45° 57'. Find the height of
The building. Round your answer to the hundredths place.

Accepted Answer

The answer of Solve the problem.
-A contractor needs to know...

Question 16

Solve the problem.
-From a distance of 1200 feet from a spotlight, the angle of elevation to a cloud base is 40°. Find the height of the cloud base to the nearest foot.

Accepted Answer

To find the height of the cloud base, use the tangent function in trigonometry: tan(θ) = opposite/adjacent. Here, θ = 40°, and the adjacent side is 1200 feet. So, height = 1200 * tan(40°) ≈ 1007 feet.

Question 17

Solve the problem.
-You are hiking and are trying to determine how far away the nearest cabin is which happens to be due north from your current position. A partner walks 160 yards due west from your position and takes a bearing on the
Cabin of N29.8°E. How far away is the cabin to the nearest yard?

Accepted Answer

Using trigonometry, the distance to the cabin can be calculated using the tangent of the angle: tan(29.8°) = opposite/adjacent = distance/160. Solving for the distance gives approximately 279 yards.

Question 18

Solve the problem.
-The chairlift at a ski resort has a vertical rise of 2600 feet. If the length of the ride is 1.2 miles, what is the average angle of inclination of the lift (to the nearest tenth of a degree)?

Accepted Answer

To find the angle of inclination, use the sine function: sin(θ) = opposite/hypotenuse. Convert 1.2 miles to feet (1.2 miles = 6336 feet). Then, sin(θ) = 2600/6336. Solving for θ gives approximately 24.2°.

Quiz 5: The Trigonometric Functions

Solve the problem. -At an altitude of 3300 ft, the engine on a small plane fails. What angle of glide is needed to reach an airport runway that is 4 miles away by land? (Round your answer to the nearest tenth of a degree.)

Solve the problem. -From a distance of 48 feet from the base of a building, the angle of elevation to the top of the building is 69°. Estimate the height of the building to the nearest foot.

Solve the problem. -A blimp is 1700 meters high in the air and measures the angles of depression to two stadiums to the west of the blimp. If those measurements are 72.4° and 27.8°, how far apart are the two stadiums to the nearest meter?

Solve the problem. -Your math class is going to test new digital clinometers by measuring the angle of elevation of a kite you will fly. The kite flies to an angle of 68.9° on 360 feet of string. Assuming that the string is taut, how high is the kite To the nearest foot?

Solve the problem. -The air speed of an airplane is 670 km/hr and its angle of climb is 6.08°. What is its ground speed (to the nearest km/hr) ?

Solve the problem. -When sitting atop a tree and looking down at his pal Joey, the angle of depression of Mack's line of sight is 35° 22'. If Joey is known to be standing 31 feet from the base of the tree, how tall is the tree (to the nearest foot) ?

Solve the problem. -A kite is currently flying at an altitude of 15 meters above the ground. If the angle of elevation from the ground to the kite is 30°, find the length of the kite string to the nearest meter.

Solve the problem. -From the top of a vertical tower, 353 feet above the the surface of the earth, the angle of depression to a dog house is 28° 51'. How far is it from the dog house to the foot of the tower? Round your answer to the hundredths Place.

Solve the problem. -A vertical antenna is mounted atop a 23-ft pole. From a point on the level ground 57 feet from the base of the pole, the antenna subtends an angle of 10.1°. Find the length of the antenna to the nearest tenth of a foot.

Solve the problem. -A 38-foot ladder is leaning against the side of a building. If the ladder makes an angle of 23° 30' with the side of the building, how far up from the ground does the ladder make contact with the building? Round your answer To the hundredths place.

Solve the problem. -From a balloon 1197 feet high, the angle of depression to the ranger headquarters is 50°21'. How far is the headquarters from a point on the ground directly below the balloon (to the nearest foot) ?

Solve the problem. -From a distance of 1200 feet from a spotlight, the angle of elevation to a cloud base is 40°. Find the height of the cloud base to the nearest foot.

Solve the problem. -The chairlift at a ski resort has a vertical rise of 2600 feet. If the length of the ride is 1.2 miles, what is the average angle of inclination of the lift (to the nearest tenth of a degree) ?