Question

The number of bacteria colonies $h$ hours after the beginning of an experiment is given by the function $C(h)=3^h-2h+20$. What does the number 20 represent in the function? A) The final rate of growth, in colonies per hour B) The initial rate of growth, in colonies per hour C) One less than the initial number of bacteria colonies D) One more than the final number of bacteria colonies $$\begin{array}{rl}& \text{ Agricultural Land as a Percent of Total Land Area, }2014\\ & \begin{array}{|lc|}\hline \text{multicolumn}1|c|\text{ Country }& \text{ Percent of Total LandArea }\\ \text{ Brazil }& 33.8\mathrm{\%}\\ \text{ Canada }& 7.2\mathrm{\%}\\ \text{ Greenland }& 0.6\mathrm{\%}\\ \text{ Latvia }& 30.1\mathrm{\%}\\ \text{ Mexico }& 54.9\mathrm{\%}\\ \text{ New Zealand }& 42.2\mathrm{\%}\\ \text{ Russian Federation }& 13.3\mathrm{\%}\\ \text{ Turkey }& 50.1\mathrm{\%}\\ \text{ United States }& 44.6\mathrm{\%}\\ \hline\end{array}\end{array}$$

Short answer

The number 20 in the function represents one less than the initial number of bacteria colonies. The correct answer is C) One less than the initial number of bacteria colonies.

Step-by-step solution

Write down the given function

The number of bacteria colonies h hours after the beginning of an experiment is given by the function: $$C(h)=3^h-2h+20$$

Calculate the initial number of bacteria colonies

To find the initial number of bacteria colonies (when h = 0), we need to substitute h = 0 into the function: $$C(0)=3^0-2(0)+20$$ $$C(0)=1-0+20$$ $$C(0)=21$$ This tells us that at the beginning of the experiment, there are 21 bacteria colonies.

Evaluate the answer choices

Now, let's evaluate the different answer choices: A) The final rate of growth, in colonies per hour: this option is not related to the initial conditions of the function, so it is not the correct answer. B) The initial rate of growth, in colonies per hour: this option refers to the initial growth rate, but it is not the number 20 in the function. The initial rate of growth is given by the derivative of the function C(h) evaluated at h=0, which is not 20. Hence, this option is incorrect. C) One less than the initial number of bacteria colonies: Since the initial number of colonies is 21, one less than this number is 20. Therefore, this option is correct. D) One more than the final number of bacteria colonies: This option is not related to the initial conditions of the function, and as such, it is not the correct answer. Based on our evaluation, the correct answer is C) One less than the initial number of bacteria colonies.

Key concepts

Algebra Functions

In this exercise, we are working with a mathematical function that describes how the number of bacteria colonies changes over time. This function is written in terms of the hour variable, denoted as $h$.
Functions like this one are used to represent relationships between quantities. In our example, the function $C(h)=3^h-2h+20$ ties the passage of time $h$ to the population of bacteria.

• Each term in the function provides different information about how the system changes.
• The term $3^h$ reflects how the quantity grows exponentially.
• The term $-2h$ presents a linear subtraction factor impacting growth.
• The constant $20$ represents an initial condition which we'll cover more later.

By understanding functions, students can analyze how varying input values affect the output thoroughly.

Exponential Growth

Exponential growth occurs when the growth rate of a value is proportional to its current size, causing the quantity to increase rapidly over time.
This concept is expressed in our function by the term $3^h$. As $h$ increases, $3^h$ grows very quickly, reflecting the nature of exponential growth.

• This is typical in biological settings, such as population studies where the number of organisms can double at regular intervals.
• Each increase in time $h$ results in multiplication of the population rather than addition.

Recognizing exponential patterns is useful for predicting outcomes based on initial settings and understanding when exponential growth may become unsustainable.
In our bacteria problem, $3^h$ is a driving force behind the changes in colony numbers post-experiment commencement.

Initial Conditions

Initial conditions help define the starting point of any calculated process, often determining the conditions where $h=0$. In this problem, the function includes an initial condition represented by the constant $20$.

• The initial condition is crucial as it represents the baseline—or starting value—of the number of bacteria colonies when no time has passed yet ($h=0$).
• By solving $C(0)=3^0-2\times 0+20$, the initial number of colonies is calculated to be 21.
• The constant 20 thus reflects a number closely tied to this initial setup.

Recognizing the importance of initial conditions allows us to place other values in context with starting figures, ensuring full comprehension of changes as time progresses.

Problem Solving Steps

Breaking down a problem into clear steps is essential for effective problem solving, especially in SAT math. Here’s how to methodically approach functions like $C(h)=3^h-2h+20$:

• **Step 1:** Clearly write down the problem's function. Identifying each part helps understand the relationship it depicts.
• **Step 2:** Evaluate what happens at the initial point, i.e., calculate $C(0)$ using the given function.
• **Step 3:** Compare this initial quantity to potential answer options. Understand what each option represents in the context of the problem.

Each step simplifies the process, ensuring that core operations are carried out correctly and thoroughly.
Using this step-by-step approach allows for clear evaluation of complex scenarios and reduces errors, which is a critical skill in both academic and real-world problem solving.