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(D) The stability of the bottle depends on the position of its center of mass. The bottle tips over when the vertical line passing through the center

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(D) The stability of the bottle depends on the position of its center of mass. The bottle tips over when the vertical line passing through the center of mass falls outside the base of the bottle.Let $h_b$ be the height of the bottle and $R$ be its radius. Let $h_s$ be the height of the shampoo inside the bottle. The center of mass of the system (bottle + shampoo) is at a height $h_{cm}$ from the base.Assuming the mass of the empty bottle is negligible compared to the shampoo,the center of mass of the shampoo is at $h_s/2$. The condition for the bottle to tip over is $	an 	heta = \frac{R}{h_{cm}}$.Since $h_{cm} = h_s/2$ and $f = h_s/h_b$,we have $h_s = f h_b$. Thus,$h_{cm} = \frac{f h_b}{2}$.Substituting this into the condition: $	an 	heta = \frac{R}{f h_b / 2} = \frac{2R}{f h_b}$.However,if we consider the mass of the bottle $(M_b)$ and the mass of the shampoo $(M_s = ho \pi R^2 h_s)$,the center of mass is $h_{cm} = \frac{M_b (h_b/2) + M_s (h_s/2)}{M_b + M_s}$.As $f$ increases,the center of mass initially lowers (increasing stability and $	heta$) and then rises (decreasing stability and $	heta$). This behavior is represented by a curve that increases to a maximum and then decreases,which matches graph $D$.

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