# Heights and distances

heights and distances are fundamental concepts that enable precise measurements of object heights and distances to inaccessible points.

**Problem 1: Measuring the Height of a Building**

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**Problem: A person stands 100 meters away from a building and observes the top of the building. The angle of elevation to the top of the building is 30 degrees. Calculate the height of the building.**

**Solution:**The height of the building (h) is the opposite side of the right triangle formed by the observer, the top of the building, and the ground.

Using the tangent function: tan(30Â°) = h / 100 meters.

h = 100 meters * tan(30Â°) = 100 meters * (âˆš3/3) â‰ˆ 57.74 meters (rounded to two decimal places).

**Problem 2: Distance between Two Objects**

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**Problem: Two people, standing at points A and B, observe a hot air balloon at an angle of elevation of 60 degrees and 45 degrees, respectively. If the distance between A and B is 200 meters, find the altitude of the hot air balloon.**

**Solution:**Let the altitude of the hot air balloon be “h” meters.

Using the tangent function for both angles: tan(60Â°) = h / x (where x is the distance from A to the balloon), and tan(45Â°) = h / (200 – x).

Solve these two equations simultaneously to find h.

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**Problem 3: Finding the Length of a Shadow**

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**Problem: A pole 12 meters tall casts a shadow of 16 meters. Find the angle of elevation of the sun.**

**Solution:**The angle of elevation (Î¸) can be found using the tangent function: tan(Î¸) = height of pole/length of shadow.

tan(Î¸) = 12 meters / 16 meters = 3/4.

Î¸ = arctan(3/4) â‰ˆ 36.87 degrees (rounded to two decimal places).

**Problem 4: Distance between Ships**

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**Problem: Two ships, A and B, are sailing towards each other. Ship A is 120 km away from Ship B. They observe each other at angles of elevation of 20 degrees and 30 degrees, respectively. Calculate the distance between the two ships.**

**Solution:**Let the distance between the two ships be “d” kilometers.

Using the tangent function for both angles: tan(20Â°) = 120 / (d – x) and tan(30Â°) = 120 / x (where x is the distance from Ship A to the point where they first observe each other).

Solve these two equations simultaneously to find d.

**Problem 5: Flying a Kite**

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**Problem: A person flying a kite has let out 150 meters of string. The angle of elevation to the kite is 40 degrees. Calculate the height of the kite above the ground.**

**Solution:**The height of the kite (h) is the opposite side of the right triangle formed by the person, the kite, and the ground.

Using the sine function: sin(40Â°) = h / 150 meters.

h = 150 meters * sin(40Â°) â‰ˆ 96.43 meters (rounded to two decimal places).

**Problem 6: Distance to a Lighthouse**

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**Problem: From a point 100 meters away from the base of a lighthouse, the angle of elevation to the top of the lighthouse is 45 degrees. Calculate the height of the lighthouse.**

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**Solution:**The height of the lighthouse (h) is the opposite side of the right triangle formed by the observer, the top of the lighthouse, and the ground.

Using the sine function: sin(45Â°) = h / 100 meters.

h = 100 meters * sin(45Â°) = 100 meters * (1/âˆš2) = 50âˆš2 meters â‰ˆ 70.71 meters (rounded to two decimal places).

**Problem 7: Angle of Depression**

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**Problem: From the top of a 60-meter-high tower, a person observes a car at an angle of depression of 30 degrees. Calculate the horizontal distance from the tower to the car.**

**Solution:**The horizontal distance (d) is the adjacent side of the right triangle formed by the person, the car, and the base of the tower.

Using the tangent function: tan(30Â°) = 60 meters / d.

d = 60 meters/tan (30Â°) = 60 meters / (âˆš3/3) â‰ˆ 103.92 meters (rounded to two decimal places).

**Problem 8: Finding the Height of a Flagpole**

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**Problem: An observer stands 30 meters away from a flagpole and observes the top of the flagpole at an angle of elevation of 60 degrees. Calculate the height of the flagpole.**

**Solution:**The height of the flagpole (h) is the opposite side of the right triangle formed by the observer, the top of the flagpole, and the ground.

Using the tangent function: tan(60Â°) = h / 30 meters.

h = 30 meters * tan(60Â°) = 30 meters * (âˆš3) â‰ˆ 51.96 meters (rounded to two decimal places).

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**Problem 9: Distance between Mountains**

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**Problem: Two mountaintops are observed from a point on the ground. The angle of elevation to the first mountaintop is 15 degrees, and the angle of elevation to the second mountaintop is 10 degrees. If the distance between the two mountaintops is 5 kilometers, calculate the distance from the observer to each mountaintop.**

**Solution:**Let the distance from the observer to the first mountaintop be “x” kilometers and the distance to the second mountaintop be “y” kilometers.

Use the tangent function for both angles: tan(15Â°) = height of first mountaintop / x and tan(10Â°) = height of second mountaintop / y.

Solve these two equations simultaneously to find x and y.

**Problem 10: Height of a Tree**

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**Problem: An observer stands 20 meters away from the base of a tree and observes the top of the tree at an angle of elevation of 45 degrees. Calculate the height of the tree.**

Solution: The height of the tree (h) is the opposite side of the right triangle formed by the observer, the top of the tree, and the ground.

Using the sine function: sin(45Â°) = h / 20 meters.

h = 20 meters * sin(45Â°) = 20 meters * (1/âˆš2) = 10âˆš2 meters â‰ˆ 14.14 meters (rounded to two decimal places).

**Problem 11: Height of a Balloon**

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**Problem: An observer on the ground observes a hot air balloon at an angle of elevation of 60 degrees. If the observer is 500 meters away from the point directly below the balloon, calculate the altitude of the balloon.**

Solution: The altitude of the balloon (h) is the opposite side of the right triangle formed by the observer, the balloon, and the ground.

Using the sine function: sin(60Â°) = h / 500 meters.

h = 500 meters * sin(60Â°) = 500 meters * (âˆš3/2) = 250âˆš3 meters â‰ˆ 433.01 meters (rounded to two decimal places).

**Problem 12: Distance between Two Buildings**

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**Problem: Two buildings are situated across from each other on a street. If the angle of elevation to the top of one building is 20 degrees and to the top of the other building is 30 degrees, and the distance between the buildings is 50 meters, calculate the height of each building.**

**Solution:**Let the height of the first building be “h1” meters and the height of the second building be “h2” meters.

Use the tangent function for both angles: tan(20Â°) = h1 / x (where x is the distance to the first building) and tan(30Â°) = h2 / (50 – x).

Solve these two equations simultaneously to find h1 and h2.

**Problem 13: Length of a Bridge**

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**Problem: An observer standing on the bank of a river sees a bridge at an angle of elevation of 45 degrees. If the observer is 40 meters away from the point directly below the bridge, calculate the length of the bridge.**

**Solution:**The length of the bridge (L) is the opposite side of the right triangle formed by the observer, the bridge, and the river.

Using the sine function: sin(45Â°) = L / 40 meters.

L = 40 meters * sin(45Â°) = 40 meters * (1/âˆš2) = 20âˆš2 meters â‰ˆ 28.28 meters (rounded to two decimal places).

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**Problem 14: Height of a Tower**

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**Problem: An observer at a distance of 15 meters from the base of a tower observes the top of the tower at an angle of elevation of 60 degrees. Calculate the height of the tower.**

**Solution:**The height of the tower (h) is the opposite side of the right triangle formed by the observer, the top of the tower, and the ground.

Using the tangent function: tan(60Â°) = h / 15 meters.

h = 15 meters * tan(60Â°) = 15 meters * (âˆš3) = 15âˆš3 meters â‰ˆ 25.98 meters (rounded to two decimal places).

**Problem 15: Distance between Two Poles**

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**Problem: Two poles are located on the same side of a road. An observer standing on the opposite side of the road observes the tops of the poles at angles of elevation of 40 degrees and 60 degrees, respectively. If the distance between the poles is 10 meters, calculate the height of each pole.**

**Solution:**Let the height of the first pole be “h1” meters and the height of the second pole be “h2” meters.

Use the tangent function for both angles: tan(40Â°) = h1 / x (where x is the distance to the first pole) and tan(60Â°) = h2 / (10 – x).

Solve these two equations simultaneously to find h1 and h2.

**Problem 16: Height of a Cliff**

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**Problem: An observer at a height of 15 meters above sea level observes a boat at an angle of depression of 30 degrees. If the boat is 50 meters away from the base of a cliff, calculate the height of the cliff.**

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**Solution:**The height of the cliff (h) is the opposite side of the right triangle formed by the observer, the boat, and the sea level.

Using the tangent function: tan(30Â°) = h / 50 meters.

h = 50 meters * tan(30Â°) = 50 meters * (1/âˆš3) â‰ˆ 28.87 meters (rounded to two decimal places).

**Problem 17: Distance between Two Observers**

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**Problem: Two observers standing at different points along a straight road observe the same hot air balloon. The angles of elevation from the two points are 20 degrees and 35 degrees, respectively. If the distance between the two observers is 100 meters, calculate the altitude of the hot air balloon.**

**Solution:**Let the altitude of the hot air balloon be “h” meters.

Use the tangent function for both angles: tan(20Â°) = h / x (where x is the distance from the first observer to the balloon), and tan(35Â°) = h / (100 – x).

Solve these two equations simultaneously to find h.

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**Problem 18: Finding the Length of a Bridge**

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**Problem: An observer stands on one side of a river and observes a bridge at an angle of elevation of 40 degrees. If the observer is 80 meters away from the point directly below the bridge, calculate the length of the bridge.**

**Solution:**The length of the bridge (L) is the opposite side of the right triangle formed by the observer, the bridge, and the river.

Using the sine function: sin(40Â°) = L / 80 meters.

L = 80 meters * sin(40Â°) â‰ˆ 51.76 meters (rounded to two decimal places).

**Problem 19: Height of a Flagpole**

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**Problem: An observer stands 40 meters away from the base of a flagpole and observes the top of the flagpole at an angle of elevation of 50 degrees. Calculate the height of the flagpole.**

**Solution:**The height of the flagpole (h) is the opposite side of the right triangle formed by the observer, the top of the flagpole, and the ground.

Using the tangent function: tan(50Â°) = h / 40 meters.

h = 40

**Problem 20: Height of a tree**

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**Problem: A person standing at a certain point observes the top of a tree at an angle of elevation of 45 degrees. If the person walks 20 meters closer to the tree and observes the top at an angle of elevation of 60 degrees, what is the height of the tree?**

**Solution:**To find the height of the tree, we can use the concept of trigonometry. Let “h” be the height of the tree in meters.

When the person is initially standing, we have tan(45 degrees) = h / d1, where d1 is the initial distance from the person to the tree.

tan(45 degrees) = h / d1.

h = d1.

When the person walks 20 meters closer to the tree, the new distance is (d1 – 20 meters), and we have tan(60 degrees) = h / (d1 – 20 meters).

âˆš3 = h / (d1 – 20).

Now, we can set these two equations equal to each other:d1 = âˆš3 * (d1 – 20).

Solving for d1:d1 = âˆš3d1 – 20âˆš3.

Bringing “âˆš3d1” to the left side:d1 – âˆš3d1 = -20âˆš3.

Combining like terms:(-âˆš3)d1 = -20âˆš3.

Dividing both sides by (-âˆš3):d1 = 20 meters.

So, the height of the tree (h) is equal to the initial distance from the person to the tree (d1), which is 20 meters.