---
title: "Reserve: quartzite apparent dips 1:9 and 1:12, 2.7 x 3.1 km"
subject: "Engineering Geology II"
chapter: 7
key: "semester_4/engineering_geology_ii:reserve-quartzite-apparent-dips-terrain"
source: https://www.bctengineer.com/q/semester_4/engineering_geology_ii/reserve-quartzite-apparent-dips-terrain
site: BCT Engineering
---

# Reserve: quartzite apparent dips 1:9 and 1:12, 2.7 x 3.1 km

Engineering Geology II · Chapter 7: Geology and Construction Materials

- Asked in: 2079 Chaitra, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/engineering_geology_ii/reserve-quartzite-apparent-dips-terrain
- Whole chapter: https://www.bctengineer.com/md/semester_4/engineering_geology_ii/chapter-7

## Question

Apparent dip amounts of quartzite bed rock along N20°E and N65°E are 1:9 and 1:12 respectively. The vertical thickness of bedrock is 105 m. Calculate the reserve of construction materials for an engineering project at 2.7 km and 3.1 km terrain.

## Short answer

From apparent dips 1:9 along N20°E and 1:12 along N65°E, true dip is 6.35° (1:9.0) towards N23.5°E. True thickness = 105 cos6.35° = 104.4 m. Surface area 2.7×3.1 km = 8.37 km² gives reserve 8.73×10⁸ m³ (0.873 km³).

## Answer

#### Step 1: true dip from the two apparent dips
For a direction of bearing $\beta$ the gradient of the bed satisfies $g_E\sin\beta + g_N\cos\beta = \tan\delta_a$ ($g_E$, $g_N$ = east and north components of the true gradient).
$$
\begin{aligned}
g_E\sin 20^\circ + g_N\cos 20^\circ &= \tfrac{1}{9} = 0.11111 \\
g_E\sin 65^\circ + g_N\cos 65^\circ &= \tfrac{1}{12} = 0.08333
\end{aligned}
$$
Solving: $g_E = 0.04434$, $g_N = 0.10211$.
$$
\tan\delta = \sqrt{0.04434^2 + 0.10211^2} = 0.1113 \Rightarrow \delta = 6.35^\circ\ (1:8.98)
$$
Dip direction $= \arctan(0.04434/0.10211) = 23.5^\circ$ (N23.5°E).

#### Step 2: true thickness
$$
t = t_v\cos\delta = 105\cos 6.35^\circ = 104.36\ \text{m}
$$

#### Step 3: volume (reserve)
Area of the terrain $= 2.7\ \text{km} \times 3.1\ \text{km} = 8.37\ \text{km}^2 = 8.37\times 10^6\ \text{m}^2$
$$
V = A \times t = 8.37\times 10^6 \times 104.36 = 8.73 \times 10^8\ \text{m}^3
$$

**Answer: true dip = 6.35° (1:9.0) towards N23.5°E; true thickness = 104.4 m; reserve of quartzite = 8.73 × 10⁸ m³ (0.873 km³).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How are the two apparent dips used?
   A. Added to get the true dip
   B. As equations gE sin β + gN cos β = tan(apparent dip), solved for the gradient components
   C. Averaged to get the true dip
   D. Subtracted to get the strike
   Answer: B. Each apparent dip is the gradient component along its bearing. (Key point: Setting up equations)

2. What true dip and direction were found?
   A. About 4.8° towards N20°E
   B. About 9.5° towards N65°E
   C. About 6.35° towards N66.5°W
   D. About 6.35° (1:9) towards N23.5°E
   Answer: D. tan δ = sqrt(0.04434^2 + 0.10211^2) = 0.1113; direction arctan(0.04434/0.10211) = 23.5°. (Key point: True dip)

3. What true thickness results for tv = 105 m?
   A. About 11.6 m
   B. About 104.4 m
   C. About 105 m
   D. About 116.9 m
   Answer: B. 105 cos 6.35° = 104.36 m. (Key point: True thickness)

4. What is the reserve over 2.7 km x 3.1 km?
   A. About 8.37 x 10^6 m³
   B. About 8.73 x 10^8 m³
   C. About 9.8 x 10^8 m³
   D. About 2.8 x 10^8 m³
   Answer: B. Area 8.37 x 10^6 m² x 104.36 m = 8.73 x 10^8 m³. (Key point: Volume)

## Close variants set in other exams

- [Reserve: quartzite apparent dips 1:9 and 1:12, 4.6 km2](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Areserve-quartzite-apparent-dips-area) — Engineering Geology II, chapter 7, asked 2×

## Related in Engineering Geology II

- [True dip from two apparent dips](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Atrue-dip-from-apparent-dips) — Engineering Geology II, chapter 6
- [Limestone reserve for cement factory](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Areserve-limestone-cement-density) — Engineering Geology II, chapter 7
- [Reserve: limestone dip 30, apparent 20 m](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Areserve-limestone-dip-30) — Engineering Geology II, chapter 7

## Related in other subjects

- [Strike of quartzite bed, dip 70° towards S20°E](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Astrike-of-quartzite-bed-dip) — Engineering Geology I, chapter 5
- [Strike of limestone bed directed N20°E, dip 45°](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Astrike-of-limestone-bed-directed) — Engineering Geology I, chapter 5