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Limestone true thickness across stream

Short answer

True thickness for sloping ground: t = w sinθ sinδ – h cosδ. With breadth w=2100 m, strike N30°W, dip 50°SW, stream south, height drop h=900 m, θ=30°, t = 2100 sin30° sin50° – 900 cos50° = 225.8 m.

A stream flows in a southerly direction across a limestone that strikes N30°W and dips 50°SW. Determine the true thickness of the limestone if the base of the limestone is exposed at an altitude of 2900 m and the top is exposed at an altitude of 2000 m. The breadth of the limestone along the stream is 2100 m.

Principle: the true thickness is the perpendicular distance between the top and bottom surfaces of the bed. When the traverse is on sloping ground it is computed from the horizontal distance along the traverse and the difference in height.

Given

Strike N30°W (330°), dip 50° SW (dip direction 240°). Stream flows south (180°). Horizontal distance (breadth) along the stream w=2100w = 2100 m. Base is at 2900 m and top at 2000 m, so the height difference h=900h = 900 m (the ground falls from base to top, i.e. downstream, which is in the dip direction).

Step 1: angle between traverse and strike

Strike line 330° (or 150°); traverse bearing 180°: θ=180∘−150∘=30∘\theta = 180^\circ - 150^\circ = 30^\circ.

Step 2: thickness for level ground

t0=wsin⁡θsin⁡δ=2100sin⁡30∘sin⁡50∘=804.35 mt_0 = w\sin\theta\sin\delta = 2100\sin 30^\circ\sin 50^\circ = 804.35\ \text{m}

Step 3: correction for slope of ground

The ground falls in the direction of dip, so the height difference reduces the thickness:

t=wsin⁡θsin⁡δ−hcos⁡δ=804.35−900cos⁡50∘=804.35−578.51=225.84 m\begin{aligned} t &= w\sin\theta\sin\delta - h\cos\delta \\ &= 804.35 - 900\cos 50^\circ \\ &= 804.35 - 578.51 = 225.84\ \text{m} \end{aligned}

Answer: true thickness of the limestone = 225.8 m (about 226 m).

✓ Answer checked by Claude Opus 5.5 (high effort)

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