Math by First Principles
Level 2 · Intermediate ← Back to lesson

Trigonometry in Measurement: Practice

12 multiple-choice questions, progressively harder.

Level 2 · Intermediate 0 / 12 answered
Question 1 of 12
  1. 1

    On level ground, an instrument 1.80 m above the base level sights the top of a vertical 16.0 m mast in the same vertical plane. The horizontal run is 25.0 m. Find the elevation angle to the nearest tenth of a degree.

    Measurements in a vertical planeSchematic, not for measuring. E is the eye above ground point F. The object BT is vertical. F and B have the same ground elevation. The dashed horizontal through E meets BT and forms a right angle. The line ET rises at θ above that horizontal. The horizontal run is 25 m, eye height is 1.8 m, and object height is labeled 16.0 m.EFBCThorizontal25 m16.0 mθEye height EF: 1.8 m
  2. 2

    An eye is 30.030.0 m above a horizontal water surface. A point on a boat deck is 2.02.0 m above that surface and is seen at depression 25.0∘25.0^\circ. All points lie in one vertical plane. Find the horizontal run to the nearest tenth of a meter.

    Measurements in a vertical planeSchematic, not for measuring. Eye E is above the base B by 30 m. Target P is 2 m above that same horizontal base level. Two dashed horizontal references are parallel. The line EP slopes down at depression angle 25° from the upper horizontal. The horizontal run is labeled d.EBP25°horizontald30 mTarget above base level: 2 m
  3. 3

    A straight ramp has sloped length 12.0 m and vertical rise 1.80 m. Find its percent gradient to the nearest tenth of a percent.

    Measurements in a vertical planeSchematic, not for measuring. A straight ramp is the sloping hypotenuse, labeled 12.0 m. The horizontal run is d and the perpendicular vertical rise is 1.80 m. The angle with the horizontal is labeled inclination.d1.80 m12.0 minclination to horizontalHorizontal run and vertical rise
  4. 4

    An observer's ground point is 3.03.0 m above a vertical tower's base level. The eye is 2.02.0 m above that ground. The horizontal run to the tower is 30.030.0 m, and elevation is 24.0∘24.0^\circ in the common vertical plane. Find tower height to the nearest tenth.

  5. 5

    From a fixed eye, the top and bottom of a vertical sign are both above the eye's horizontal, at elevations 39.0∘39.0^\circ and 27.0∘27.0^\circ respectively. Their common horizontal run is 18.018.0 m. Find the sign's vertical length to the nearest tenth.

    Measurements in a vertical planeSchematic, not for measuring. Both endpoints of vertical segment TB lie above eye E's horizontal. The elevation to top T is 39 degrees and to base B is 27 degrees. Both sights have horizontal run 18 meters. Blue solid and red dashed lines reach T and B respectively. Segment length is labeled ell.ETBℓhorizontalElevation to T: 39°; to B: 27°Horizontal run: 18 m
  6. 6

    Two equal-height instruments stand on the same straight horizontal line through a vertical tower base, with one station nearer the base. Both sight the same top above eye level. Which angle pair is compatible with this order?

  7. 7

    From one station, an observer measures only elevation angles to the top and base of a vertical segment, both above the horizontal. No distance or length is known. Does this determine the segment’s length?

  8. 8

    A report uses eye height 1.651.65 m, elevation 36∘36^\circ above horizontal, and a 3232 m tape distance measured along sloping ground to estimate a vertical tower's height as 1.65+32tan⁡36∘1.65+32\tan36^\circ. No ground inclination or base-level difference is recorded. What is the justified conclusion?

  9. 9

    A vertical target is 17.0 m high above level ground. An eye 1.4 m above the same ground is horizontally 23.0 m away, in the same vertical plane. Find the top’s elevation angle to the nearest tenth of a degree.

    Measurements in a vertical planeSchematic, not for measuring. E is the eye above ground point F. The object BT is vertical. F and B have the same ground elevation. The dashed horizontal through E meets BT and forms a right angle. The line ET rises at θ above that horizontal. The horizontal run is 23 m, eye height is 1.4 m, and object height is labeled 17.0 m.EFBCThorizontal23 m17.0 mθEye height EF: 1.4 m
  10. 10

    A straight ramp is 5.05.0 m long along its surface and rises at 12.0∘12.0^\circ to the horizontal. Find its horizontal run to the nearest tenth of a meter.

    Measurements in a vertical planeSchematic, not for measuring. A straight ramp is the sloping hypotenuse, labeled 5.0 m. The horizontal run is d and the perpendicular vertical rise is r. The angle with the horizontal is labeled 12°.dr5.0 m12°Horizontal run and vertical rise
  11. 11

    An eye sees the top of a vertical tower at elevation 27.0∘27.0^\circ and its base at depression 18.0∘18.0^\circ. Both angles use the eye's horizontal in a common vertical plane. The horizontal run is 35.035.0 m. Find whole tower height to the nearest tenth.

    Measurements in a vertical planeSchematic, not for measuring. Eye E sees the top T above its horizontal at elevation 27 degrees and the base B below its horizontal at depression 18 degrees. BT is vertical, and the horizontal run from E to BT is 35 m. The total vertical height BT is H.EBT27°18°horizontal35 mH
  12. 12

    An observer stands on ground 2.02.0 m below the base level of a vertical tower. The eye is 1.71.7 m above the observer's ground. Horizontal run is 40.040.0 m and elevation is 36.0∘36.0^\circ in one vertical plane. Find tower height to the nearest tenth.

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