Coordinates and trigonometry · Ages 14–15

Gradient and inclination worksheet

Part of: Trigonometry

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Coordinates and trigonometry · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.A slope rises 6 m in 33 m across. To the nearest 0.1°, what is its inclination θ?
Answer:
2.A slope runs 30 m across and rises 5 m. What is its gradient, as a fraction in its lowest terms?
Answer in simplest form
Answer:
3.A slope rises 10 m in 37 m across. To the nearest 0.1°, what is its inclination θ?
Answer:
4.A slope has a gradient of 1 : 27. To the nearest 0.1°, what is its inclination?
Answer:
5.Someone walks 190 m up a slope inclined at 19°. Correct to 1 decimal place, how many metres higher are they?
Answer:
6.Someone walks 180 m up a slope inclined at 30°. Correct to 1 decimal place, how many metres higher are they?
Answer:
7.A slope has a gradient of 1 : 5. To the nearest 0.1°, what is its inclination?
Answer:
8.A slope with a gradient of 1 : 19 runs 240 m across. Correct to 1 decimal place, how long is it along the slope, in m?
Answer:
9.A road sign shows the road ahead has a gradient of 18%. To the nearest 0.1°, what is its inclination?
Answer:

10.A slope has a gradient of 1 : 2. Going 100 m across, how much does it rise?
- A. 1003 m
- B. 150 m
- C. 50 m
- D. 200 m
Choose the right answer
Answer:
11.A slope with a gradient of 1 : 6 is steeper than one inclined at 12°.
True or false?
Answer:
12.A slope with a gradient of 1 : 3 is steeper than one inclined at 4°.
True or false?
Answer:

13.A cable car line is 190 m long, measured along it, at an inclination of 18°. Correct to 1 decimal place, how far across does it run on a map, in m?
Working:
Answer:
14.A wheelchair ramp may be no steeper than 1 : 20. To rise 20 cm, how many metres across does it need at the least?
Working:
Answer:
15.A road sign says the road ahead has a gradient of 9%. To the nearest 0.1°, its inclination is °; walking 650 m along it, correct to 1 decimal place, you climb m.
Fill in each blank
Working:
Answer:
16.On a map the foot of a hill is 1500 m across from its top, and the top is 80 m higher. To the nearest 0.1°, what is the average inclination?
Working:
Answer:
17.A cable car line is 210 m long, measured along it, at an inclination of 9°. Correct to 1 decimal place, how far across does it run on a map, in m?
Working:
Answer:
18.A wheelchair ramp may be no steeper than 1 : 12. To rise 85 cm, how many metres across does it need at the least?
Working:
Answer:

Show the answers and working

Gradient and inclination worksheet: Answers and working

1. 10.3°
tan θ = gradient = 633, so θ = tan⁻¹(633) ≈ 10.3°.
2. 16
Gradient = vertical ÷ horizontal = 530 = 16.
3. 15.1°
tan θ = gradient = 1037, so θ = tan⁻¹(1037) ≈ 15.1°.
4. 2.1°
tan θ = 127, so θ = tan⁻¹(127) ≈ 2.1°.
5. 61.9 m
The distance along the slope is the hypotenuse: the rise = 190 × sin 19° ≈ 61.9 m.
6. 90 m
The distance along the slope is the hypotenuse: the rise = 180 × sin 30° ≈ 90 m.
7. 11.3°
tan θ = 15, so θ = tan⁻¹(15) ≈ 11.3°.
8. 240.3 m
The inclination θ = tan⁻¹(119); along the slope = 240 ÷ cos θ ≈ 240.3 m. Or by Pythagoras: the rise is 24019, the length 240² + (24019)².
9. 10.2°
18% rises 18 m for every 100 m across: tan θ = 18100, θ ≈ 10.2°.

10. C. 50 m
Gradient = rise ÷ across = 12: the rise = 100 ÷ 2 = 50 m.
11. False
A gradient of 1 : 6 is an inclination of tan⁻¹(16) ≈ 9.5°; compare it with 12°.
12. True
A gradient of 1 : 3 is an inclination of tan⁻¹(13) ≈ 18.4°; compare it with 4°.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 180.7 m
A map shows the distance across, and the line is the hypotenuse: 190 × cos 18° ≈ 180.7 m.
14. 4 m
1 : 20 needs 20 across for every 1 up: 20 × 20 cm = 4 m.
15. 5.1, 58.3
tan θ = 9100, θ ≈ 5.1°; the climb = 650 × sin θ ≈ 58.3 m (with θ unrounded).
16. 3.1°
tan θ = 801500, θ ≈ 3.1°.
17. 207.4 m
A map shows the distance across, and the line is the hypotenuse: 210 × cos 9° ≈ 207.4 m.
18. 10.2 m
1 : 12 needs 12 across for every 1 up: 85 × 12 cm = 10.2 m.

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