Write the Density index formula.

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Option C:

(e_{m a x} - e) / (e_{m a x} - e_{m i n})

Explanation & Concepts: The Density Index ($I_D$), also known as the Relative Density ($D_r$), is a measure used to quantify the degree of compactness of cohesionless (granular) soils. It expresses the current void ratio of the soil relative to its loosest and densest possible states. Working Principle: In cohesionless soils, the stability and shear strength depend on the arrangement of particles. $I_D$ compares the existing void ratio ($e$) with the minimum ($e_{min}$) and maximum ($e_{max}$) void ratios. A value of 0 implies the soil is at its loosest state, while a value of 1 (or 100%) implies the soil is at its densest state. Key Formulas: $I_D = \frac{e_{max} - e}{e_{max} - e_{min}}$ — Calculation of Density Index; $I_D = \frac{\gamma_{d} - \gamma_{dmin}}{\gamma_{dmax} - \gamma_{dmin}} imes \frac{\gamma_{dmax}}{\gamma_{d}}$ — Alternate formula using dry unit weights Key Points: Used specifically for cohesionless, granular soils (sands, gravels).; Range of $I_D$: $0 \le I_D \le 1$.; Classifies soil as very loose ($85\%$). Advantages: Provides a reliable indicator of the strength and compressibility of granular deposits.; Essential for evaluating the liquefaction potential of sands under seismic loading. Disadvantages: Difficulty in accurately determining $e_{min}$ and $e_{max}$ in laboratory settings. Applications: Foundation engineering design for granular soil sites.; Assessing stability of slopes and embankments composed of granular materials. Additional Info: In terms of unit weights, $I_D$ can be expressed via dry density $\gamma_d$, where $\gamma_{dmax}$ and $\gamma_{dmin}$ are the maximum and minimum dry unit weights respectively.. Option A is inverted, Option B uses incorrect variables in the denominator, and Option D is the reciprocal. Answer: C is correct — The density index is defined as the ratio of the difference between the maximum void ratio and the natural void ratio to the range of void ratios (difference between maximum and minimum). Exam Tip: Always verify if the question asks for void ratio ($e$) or dry unit weight ($\gamma_d$); they have inverse relationships in the $I_D$ formula structure.

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Option A: (emax-emin)/(emin-emax)

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Option B: (emax-e)/(emin-e)

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Option D: (emax-emin)/(emax-e)

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