Real-World Modeling with Angles of Depression/Elevation

  • 10th Grade
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| Attempts: 11 | Questions: 20 | Updated: Dec 11, 2025
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1) A sign that is 18 m tall casts a 40 m shadow. What is the sun's angle of elevation (nearest degree)?

Explanation

Use tangent with height 18 m and shadow 40 m:

tan θ = opposite / adjacent = 18 / 40 = 0.45

Now find θ:

θ = arctan(0.45) ≈ 24°

Hence, the sun’s angle of elevation is approximately 24°.

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About This Quiz
Real-world Modeling With Angles Of Depression/Elevation - Quiz

Apply the method to lifelike contexts—lighthouses, cranes, ramps, drones, and roads. You’ll read what the numbers mean (slope, visibility, clearance), decide whether you need horizontal, vertical, or slant distance, and communicate results with correct units and sensible rounding. This is where modeling meets measurement.

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2) A lighthouse is 38 m tall. The angle of depression to a ship is 16°. How far is the ship from the base horizontally (nearest meter)?

Explanation

Use the tangent ratio, with 38 m vertical and x horizontal:

tan 16° = 38 / x

Solve for x:

x = 38 / tan 16° ≈ 38 / 0.2867 ≈ 132.5

Rounded to the nearest meter:

x ≈ 133 m.

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3) A crane operator is 24 m above ground level. The angle of depression to a pallet is 34°. Find the horizontal distance to the pallet (nearest meter).

Explanation

Use tangent with a vertical height of 24 m and horizontal distance x:

tan 34° = 24 / x

Solve for x:

x = 24 / tan 34° ≈ 24 / 0.6745 ≈ 35.6

Rounded to the nearest meter:

x ≈ 36 m.

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4) A security post is 14 ft high. The angle of depression to a package on the ground is 33°. What is the horizontal distance to the package (nearest foot)?

Explanation

Use tangent with 14 ft vertical and horizontal distance x:

tan 33° = 14 / x

Solve for x:

x = 14 / tan 33° ≈ 14 / 0.6494 ≈ 21.6

Rounded to the nearest foot:

x ≈ 22 ft.

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5) A bridge is 32 m above the river. A boat is 220 m away horizontally. What is the angle of depression from the bridge to the boat (nearest degree)?

Explanation

We know the vertical drop is 32 m and horizontal distance is 220 m.

Use tangent:

tan θ = opposite / adjacent = 32 / 220 ≈ 0.1455

Find θ:

θ = arctan(0.1455) ≈ 8.3°

Rounded to the nearest degree:

θ ≈ 8°.

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6) An observer on a dune 11 m high looks down at a flag with an angle of depression of 17°. What is the line-of-sight distance to the flag (nearest meter)?

Explanation

Let s be the line-of-sight distance to the flag.

The vertical side is 11 m, and the angle is 17°.

Use sine:

sin 17° = 11 / s

Solve for s:

s = 11 / sin 17° ≈ 11 / 0.2924 ≈ 37.6

Rounded to the nearest meter:

s ≈ 38 m.

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7) From the roof of a 42 m building, the angle of depression to a truck is 19°. What is the horizontal distance to the truck (nearest meter)?

Explanation

We use the tangent ratio, with 42 m as the vertical side and x as the horizontal distance:

tan 19° = 42 / x

Solve for x:

x = 42 / tan 19° ≈ 42 / 0.3443 ≈ 121.99

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8) A drone flies at 320 ft altitude. The angle of depression to a landing pad is 14°. What is the line-of-sight distance to the pad (nearest foot)?

Explanation

Let s be the line-of-sight distance from the drone to the pad.

The vertical side is 320 ft, and the angle at the drone is 14°.

Use sine:

sin 14° = 320 / s

Solve for s:

s = 320 / sin 14° ≈ 320 / 0.2419 ≈ 1322.2

Submit
9) A lookout on a 72 m cliff sees a boat at an angle of depression of 11°. Find the horizontal distance to the boat (nearest meter).

Explanation

Use the tangent ratio, with 72 m vertical and x horizontal:

tan 11° = 72 / x

Solve for x:

x = 72 / tan 11° ≈ 72 / 0.1944 ≈ 370.5

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10) A surveyor is 90 m from a tower. The angle of elevation to the top is 31°. Estimate the tower's height (nearest meter).

Explanation

The surveyor is 90 m from the tower; use tangent to find the height.

tan 31° = height / 90

Solve for height:

height = 90 × tan 31° ≈ 90 × 0.6009 ≈ 54.1

Rounded to the nearest meter:

height ≈ 54 m.

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11) A hiker stands on a ledge 27 m above a lake. The angle of depression to a canoe is 22°. What is the line-of-sight distance to the canoe (nearest meter)?

Explanation

Let s be the line-of-sight distance from the hiker to the canoe.

The vertical side is 27 m, angle at the hiker is 22°.

Use sine:

sin 22° = 27 / s

Solve for s:

s = 27 / sin 22° ≈ 27 / 0.3746 ≈ 72.1

Rounded to the nearest meter:

s ≈ 72 m.

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12) A walkway rises 1.2 m over a horizontal run of 9.0 m. What is the angle of elevation of the walkway (nearest degree)?

Explanation

Use tangent with rise 1.2 m and run 9.0 m:

tan θ = 1.2 / 9.0 = 0.1333

Now find θ:

θ = arctan(0.1333) ≈ 7.6°

Rounded to the nearest degree:

θ ≈ 8°.

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13) A person stands on a pier 7 m above the water. The angle of depression to a buoy is 29°. What is the horizontal distance to the buoy (nearest tenth of a meter)?

Explanation

The vertical drop is 7 m and horizontal distance is x.

Use tangent:

tan 29° = 7 / x

Solve for x:

x = 7 / tan 29° ≈ 7 / 0.5543 ≈ 12.63

Rounded to the nearest tenth:

x ≈ 12.6 m.

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14) A balloon is 150 m above a field. A marker on the field is 520 m away horizontally. What is the angle of depression from the balloon to the marker (nearest degree)?

Explanation

tan θ = 150 / 520 ≈ 0.2885 ⇒ θ ≈ 16°.

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15) A camera mounted 12 ft high views a point on the ground at a 27° angle of depression. What is the line-of-sight distance to the point (nearest foot)?

Explanation

Let s be the line-of-sight distance from the camera to the point.

The vertical side is 12 ft, and the angle is 27°.

Use sine:

sin 27° = 12 / s

Solve for s:

s = 12 / sin 27° ≈ 12 / 0.4540 ≈ 26.4

Rounded to the nearest foot:

s ≈ 26 ft.

Submit
16) A paraglider is at an altitude of 410 m. The line-of-sight distance to a landing zone is 900 m. What is the angle of depression to the landing zone (nearest degree)?

Explanation

Let θ be the angle of depression.

The vertical side is 410 m, and the line-of-sight distance is 900 m.

Use sine:

sin θ = opposite / hypotenuse = 410 / 900 ≈ 0.4556

Now find θ:

θ = arcsin(0.4556) ≈ 27°

Hence, the angle of depression is approximately 27°.

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17) When the sun's angle of elevation is 41°, a tree casts an 18 m shadow. How tall is the tree (nearest meter)?

Explanation

Use tangent with height h and shadow 18 m:

tan 41° = height / 18

Solve for height:

height = 18 × tan 41° ≈ 18 × 0.8693 ≈ 15.65

Rounded to the nearest meter:

height ≈ 16 m.

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18) A quadcopter is 95 m above the ground. The angle of depression to a target is 26°. What is the line-of-sight distance to the target (nearest meter)?

Explanation

Let s be the line-of-sight distance.

The vertical side is 95 m, and the angle at the quadcopter is 26°.

Use sine:

sin 26° = 95 / s

Solve for s:

s = 95 / sin 26° ≈ 95 / 0.4384 ≈ 216.8

Rounded to the nearest meter:

s ≈ 217 m.

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19) A road descends at a constant 8° over a horizontal distance of 600 m. What is the change in elevation (nearest meter)?

Explanation

Use the tangent ratio with horizontal distance 600 m:

drop = 600 × tan 8°

Compute:

tan 8° ≈ 0.1405

drop ≈ 600 × 0.1405 ≈ 84.3

Rounded to the nearest meter:

drop ≈ 84 m.

Submit
20) An observation deck is 55 m high. The angle of depression to a van in the parking lot is 13°. Find the horizontal distance to the van (nearest meter).

Explanation

Use tangent with tower height h and horizontal distance x:

tan 13° = 55 / x

Solve for x:

x = 55 / tan 13° ≈ 55 / 0.2309 ≈ 238.3

Rounded to the nearest meter:

x ≈ 238 m.

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A sign that is 18 m tall casts a 40 m shadow. What is the sun's angle...
A lighthouse is 38 m tall. The angle of depression to a ship is...
A crane operator is 24 m above ground level. The angle of depression...
A security post is 14 ft high. The angle of depression to a package on...
A bridge is 32 m above the river. A boat is 220 m away horizontally....
An observer on a dune 11 m high looks down at a flag with an angle of...
From the roof of a 42 m building, the angle of depression to a truck...
A drone flies at 320 ft altitude. The angle of depression to a landing...
A lookout on a 72 m cliff sees a boat at an angle of depression of...
A surveyor is 90 m from a tower. The angle of elevation to the top is...
A hiker stands on a ledge 27 m above a lake. The angle of depression...
A walkway rises 1.2 m over a horizontal run of 9.0 m. What is the...
A person stands on a pier 7 m above the water. The angle of depression...
A balloon is 150 m above a field. A marker on the field is 520 m away...
A camera mounted 12 ft high views a point on the ground at a 27°...
A paraglider is at an altitude of 410 m. The line-of-sight distance to...
When the sun's angle of elevation is 41°, a tree casts an 18 m...
A quadcopter is 95 m above the ground. The angle of depression to a...
A road descends at a constant 8° over a horizontal distance of 600...
An observation deck is 55 m high. The angle of depression to a van in...
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