Question

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

Short answer

To answer based on the description, (a) the density of water at 10°C is determined by substituting \(t = 10\) into the given equation, (b) to find the temperature for a certain density, we need to solve the equation for \(t\), and (c) the maximum density can be found using estimation, graphical, and calculus-based methods.

Step-by-step solution

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, \(d(10) = \frac{0.99984+(1.6945 \times 10^{-2}*10)-(7.987 \times 10^{-6} *10^{2})}{1+(1.6880 \times 10^{-2} *10)}\).

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.

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.

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.

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.

Key concepts

Celsius Temperature

Understanding Celsius temperature is crucial for this exercise as it directly relates to the changes in water density. The Celsius scale, denoted as °C, is a metric scale for temperature widely used around the world.
Its freezing and boiling points of water are at 0°C and 100°C respectively under standard atmospheric pressure. This scale makes it intuitive to understand temperature changes during physical phenomena.

In this exercise, we investigate how the density of water varies with temperature in a narrow range from 0°C to 20°C. Generally, the density of a substance decreases as temperature increases because the substance's volume expands. But water has unique properties that result in non-linear behaviour.
Knowing how to express and interpret phenomena in Celsius allows us to easily engage with and understand experimental results and scientific insights.

Numerical Method

The numerical method is applied to solve for the unknown temperature at which water attains a given density. This problem is not straightforwardly solved through simple algebra and necessitates approximation techniques.

In our context, it's about setting the given density equation equal to a particular density value, such as 0.99860 g/cm³, and then finding the corresponding temperature using a computational approach.
Numerical methods often involve iterative procedures.

• Advantages: These methods can handle complex equations that escape simple algebraic solutions.
• Current Usage: We frequently rely on tools like computer algorithms to implement numerical methods efficiently, especially when determining results across variable ranges.
• Outcome: Through this method, we can approximate the temperature desired for the specific density value, making numerical methods a powerful tool in scientific calculations.

Graphical Method

Graphs provide a visual representation of mathematical equations and can be especially helpful in identifying trends and relationships. In our exercise, the graphical method involves plotting the density equation over the set temperature range.
This visual inspection allows us to define at what point the density is maximized by observing the highest peak on the graph.

Graphical methods are valuable because they offer:

• Intuitive Understanding: Making it easier to visually grasp where maximum or minimum points lie.
• Pattern Identification: Identifying trends quickly, such as whether the density increases or decreases with changes in temperature.
• Immediate Insights: Simplicity in spotting outliers or unusual behavior.

Overall, graphical methods are an excellent way to complement and verify computational results and provide a tangible view of abstract mathematical concepts.

Differential Calculus

Differential calculus is a branch of mathematics focused on rates of change and slopes of curves. It can be used to find local maxima and minima of functions, like determining maximum density in our exercise.

In this situation, taking the derivative of the density function in relation to temperature ( \( rac{d}{dt} ext{density} \) ) helps find when the slope of the curve hits zero.
This point indicates where the density function has reached a maximum or minimum. Once the derivative is set to zero and solved for t, it provides the temperature where maximum density occurs.

Key aspects of using differential calculus:

• Precision: Allows us to accurately pinpoint the temperature for maximum density within our specified range.
• Theoretical Confirmations: Proves theoretical assumptions about relationships between variables mathematically.
• Wide Applicability: Beyond finding maximum density, it's employed across scientific and engineering fields for optimizing numerous systems and models.

Hence, differential calculus is paramount for providing scientific and mathematically sound answers in scenarios like our exercise.