Problem 32 - Algebra Form and Function

Question

The growth rate of a colony of bacteria at temperature \(T^{\circ} \mathrm{F}\) is \(P(T)\). The Fahrenheit temperature \(T\) for \(H^{\circ} \mathrm{C}\) is $$ T=\frac{9}{5} \cdot H+32 $$ Find an expression for \(Q(H),\) the growth rate as a function of \(H\)

Step-by-Step Solution

Answer

Question: Write an expression for Q(H), the growth rate of a colony of bacteria as a function of Celsius temperature, H, given that the growth rate is P(T) as a function of temperature in Fahrenheit. Answer: Q(H) = P(5/9 (T-32))

Step 1: Solve the given equation for H in terms of T

We are given the equation to convert temperature from Celsius to Fahrenheit: $$ T = \frac{9}{5} \cdot H + 32 $$ We need to solve this equation for H in terms of T. To do this, first subtract 32 from both sides and then multiply both sides of the resulting equation by \(\frac{5}{9}\) to isolate H. $$ T - 32 = \frac{9}{5} \cdot H $$ $$ H = \frac{5}{9} (T-32) $$

Step 2: Substitute the expression for H in terms of T into the growth rate function P(T)

Now that we have an expression for H in terms of T, we can substitute this expression into our given growth rate function P(T) to find Q(H). Since H is expressed in terms of T, replacing T with the given expression will give us an expression for Q(H). $$ Q(H) = P\left(\frac{5}{9} (T-32)\right) $$ Since the growth rate function, P(T), is not given, this is our final expression for Q(H). We have successfully expressed the growth rate of the colony of bacteria as a function of temperature in Celsius, H.

Key Concepts

Fahrenheit to Celsius

The Fahrenheit and Celsius temperature scales are two of the most commonly used temperature measurements. To convert temperatures from Fahrenheit to Celsius, we use a specific mathematical formula.

• The formula to convert Fahrenheit ((T)) to Celsius ((H)) is: \[ H = \frac{5}{9} (T - 32) \]
• This formula arises from the linear relationship between the Fahrenheit and Celsius scales, where the freezing point of water is 32°F (0°C) and the boiling point is 212°F (100°C).

A simple method to use the formula:

• Subtract 32 from the Fahrenheit temperature.
• Multiply the result by \(\frac{5}{9}\).

This conversion can be important for many scientific calculations, including those in biology where temperature often influences experimental conditions.

Growth Rate Function

Understanding how a growth rate function works is essential in biology and many other fields. When studying populations, such as bacteria colonies, the growth rate function, represented as (P(T)) , may describe how the population size changes over time relative to temperature.

• A growth rate function typically includes variables that affect the growth of an organism or colony, such as temperature, available resources, or time.
• In our example, (P(T)) specifically depends on temperature measured in Fahrenheit.

To convert the growth rate dependency from Fahrenheit to Celsius, we need to express the function in terms of (H) , the Celsius temperature. This involves substituting the converted temperature into the original growth equation. While the exact formulation of (P(T)) is hypothetical in this case, understanding the process is crucial for practical applications.

Mathematical Expressions

Mathematical expressions are used to represent complex relationships in a compact, symbolic form. In the given exercise, we use such expressions to convert temperature and evaluate growth rate functions.

• The conversion of temperature uses the expression \( H = \frac{5}{9} (T - 32) \), demonstrating a linear transformation from Fahrenheit to Celsius.
• Applying this transformation in a biological context, we substitute it into the growth rate function (P(T)) to create (Q(H)), which is an expression of growth rate in terms of Celsius temperature.

When working with mathematical expressions:

• Identify the variables and constants involved.
• Understand their relationships and how changing one affects the others.
• Substitute expressions carefully and check for logical consistency.

This approach allows us to model real-world phenomena with precision and clarity.