16.1 Engineering Mathematics for Safe, Efficient Mine Decisions

Applied mathematics is not a separate exam island. It is the language of ventilation, blasting, slope stability, surveying, finance, production, processing, and environmental control. Most avoidable errors come from wrong units, hidden assumptions, sign conventions, premature rounding, or a result that was never checked against physical reality.

Dimensional Discipline

Write units beside every value and convert before substitution. One kilopascal is one kilonewton per square metre; one cubic metre per second is 1,000 litres per second; one percent is a fraction of 0.01. If a formula for energy yields metres per second, the setup is wrong.

For pumping water at flow $Q$, head $H$, density $\rho$, gravity $g$, and total efficiency $\eta$:

$P=\frac{\rho gQH}{\eta}$

At $Q=0.12$ m3/s, $H=80$ m, $\rho=1000$ kg/m3, and $\eta=0.75$:

$P=\frac{1000(9.81)(0.12)(80)}{0.75}=125{,}568\ W \approx126\ kW$

The water receives about 94 kW of hydraulic power; dividing by 75% total efficiency gives about 126 kW input power. That result excludes starting duty and design margin and should not lead automatically to a 126-kW nameplate motor.

Algebra, Ratios and Proportion

Rearrange symbolically before inserting numbers. This reduces repeated arithmetic and reveals proportional behavior. Atkinson's ventilation relation $H=RQ^2$ shows that doubling airflow requires four times pressure at unchanged resistance. Power $HQ$ then grows approximately with the cube of flow.

Percent change differs from percentage-point change. Recovery increasing from 80% to 84% rises 4 percentage points but 5% relative to its original value. In grade and recovery calculations, convert all percentages consistently or retain all in percent form where ratios cancel.

Trigonometry and Vectors

For a ramp of horizontal run 500 m and rise 50 m, grade is $50/500=10%$ and angle is $\tan^{-1}(0.10)=5.71$ degrees. Do not confuse grade percent with degrees.

A force or survey displacement has magnitude and direction. Resolve a vector $F$ at angle $\theta$ into:

$F_x=F\cos\theta, \qquad F_y=F\sin\theta$

State the reference direction and quadrant. Calculator inverse-tangent output alone can place a bearing in the wrong quadrant.

Geometry and Volumes

Approximate a bench volume using average end area:

$V=L\frac{A_1+A_2}{2}$

For irregular surfaces, use surveyed triangulated surfaces or grid/block volumes appropriate to the required accuracy. Swell and compaction change bulk volume but not dry mass; apply density and moisture on the correct basis.

Rates, Derivatives and Accumulation

A rate is a change per unit time. If pit inflow is $q(t)$, accumulated water is $V=\int q(t)dt$. A constant 40 L/s over six hours yields $40 \times 3600 \times 6=864{,}000$ L, or 864 m3. A pump nominally rated 50 L/s provides only 10 L/s net drawdown before losses and variability, so the sump and response time matter.

Production curves can be differentiated to estimate instantaneous rate or integrated to estimate cumulative tonnes. Numerical integration, such as trapezoidal summation, is used when measurements vary by interval.

Statistics for Decisions

• Mean: arithmetic center, sensitive to outliers.
• Median: middle value, robust for skewed data.
• Standard deviation: dispersion around the mean.
• Bias: systematic difference from a reference.
• Precision: repeatability, which can be high even when biased.
• Confidence interval: range reflecting sampling uncertainty under stated assumptions.

Ten duplicate assays that agree closely show precision; they do not prove accuracy if both share calibration bias. Sample support and spatial correlation also matter: 100 assays from one ore shoot are not equivalent to 100 distributed holes.

Final Answer Protocol

1. sketch or time-line the problem;
2. define symbols and units;
3. state assumptions and governing equation;
4. solve with guard digits;
5. convert to requested units and precision; and
6. test sign, magnitude, limiting case, and operational meaning.

A mathematically correct negative fan flow or 140% recovery signals a definition or data error. Engineering mathematics promotes safety when it exposes such impossible results before they enter a design.

Probability and Expected Loss

Risk calculations combine likelihood and consequence only when both are defined consistently. If a pump failure has estimated annual probability 0.08 and direct consequence PHP 12 million, simple expected annual loss is PHP 0.96 million. This does not make a catastrophic event acceptable: life-safety, legal, environmental, and tail-risk criteria can impose hard controls beyond expected value. Avoid adding dependent event probabilities as if independent. Use event trees or conditional probability when one failure changes another's likelihood.

Interpolation Check

Linear interpolation between measured points assumes approximately linear change within the interval. Do not extrapolate far beyond the observed range without a physical model. In calibration, a result beyond the highest standard may require dilution and remeasurement, not extended use of the fitted line.