Between 0^∘ C and 30^∘ C, the volume V( in cubic centimeters) of 1 kg of water at a temperature

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Between 0^∘ C and 30^∘ C, the volume V( in cubic...

Between $0^{\circ} \mathrm{C}$ and $30^{\circ} \mathrm{C},$ the volume $V($ in cubic centimeters) of 1 $\mathrm{kg}$ of water at a temperature $T$ is given approximately by the formula $$ V=999.87-0.06426 T+0.0085043 T^{2}-0.0006679 T^{3} $$ Find the temperature at which water has its maximum density.

Between $0^{\circ} \mathrm{C}$ and $30^{\circ} \mathrm{C},$ the volume $V($ in cubic centimeters)
of 1 $\mathrm{kg}$ of water at a temperature $T$ is given approximately by
the formula
$$
V=999.87-0.06426 T+0.0085043 T^{2}-0.0006679 T^{3}
$$
Find the temperature at which water has its maximum
density.

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Key Concepts

Relationship Between Density and Volume

Density is defined as mass divided by volume, so for a fixed mass, maximizing the density of water is equivalent to minimizing its volume. Recognizing this relationship is crucial when the problem asks for the temperature corresponding to the maximum density, leading to the strategy of finding the minimum point of the volume function by using calculus techniques.

Critical Points

Critical points refer to values in the domain of a function where its derivative is zero or undefined, which are potential candidates for local maxima or minima. In this context, finding the critical point involves differentiating the volume expression with respect to temperature and setting the derivative equal to zero to locate the temperature where the density of water is maximized (or equivalently, the volume is minimized).

Polynomial Functions

A polynomial function is an expression consisting of variables raised to non-negative integer powers and associated coefficients. The given volume equation is a cubic polynomial in temperature, and understanding the behavior of polynomial functions, including their differentiability and the nature of their graphs, is essential in analyzing and solving for extrema such as the temperature of maximum density.

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