The following equation can be used to relate the density of liquid water to Celsius temperature in the range from \(0^{\circ} \mathrm{C}\) to about \(20^{\circ} \mathrm{C}:\) $$d\left(\mathrm{g} / \mathrm{cm}^{3}\right)=\frac{0.99984+\left(1.6945 \times 10^{-2} t\right)-\left(7.987 \times 10^{-6} t^{2}\right)}{1+\left(1.6880 \times 10^{-2} t\right)}$$ (a) To four significant figures, determine the density of water at \(10^{\circ} \mathrm{C}\). (b) At what temperature does water have a density of \(0.99860 \mathrm{g} / \mathrm{cm}^{3} ?\) (c) In the following ways, show that the density passes through a maximum somewhere in the temperature range to which the equation applies. (i) by estimation (ii) by a graphical method (iii) by a method based on differential calculus

Step by step solution

01

Density at 10°C

We're asked to determine the density at 10°C, this simply involves substituting \(t = 10\) into the given equation. So, \(d(10) = \frac{0.99984+(1.6945 \times 10^{-2}*10)-(7.987 \times 10^{-6} *10^{2})}{1+(1.6880 \times 10^{-2} *10)}\).

02

Temperature for Certain Density

We're asked to find the temperature that yields a density of \(0.99860 \: \mathrm{g} / \mathrm{cm}^{3}\). This involves making the equation equal to \(0.99860 \) and then solving for \(t\). You'll have to use a numerical method for this as the equation won't be solvable by simple algebraic manipulations.

03

Estimation Method for Maximum Density

Maximum density by estimation involves providing a range of temperatures and determining where the density has the largest value. For simplicity, take every 5°C within the given range and compute the density. The temperature at which the largest density occurs is your estimated maximum.

04

Graphical Method for Maximum Density

For the graphical method, you'll need to plot the given equation over the temperature range. Once we do this, observe the graph for the temperature at which the density is at a maximum.

05

Calculus Based Method for Maximum Density

Determine the local maxima using calculus by first calculating the derivative of density function with respect to \(t\) (which should give a quadratic equation). Then, set this equal to zero and solve for \(t\) (the temperature)). The appropriate root (within the given temperature range) will be the temperature of maximum density.

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