Question

You plant a sunflower seedling in your garden. The height \(h\) (in centimeters) of the seedling after \(t\) weeks can be modeled by the logistic function $$ h(t)=\frac{256}{1+13 e^{-0.65 t}} . $$ a. Find the time it takes the sunflower seedling to reach a height of 200 centimeters. b. Use a graphing calculator to graph the function. Interpret the meaning of the asymptote in the context of this situation.

Short answer

a. The time it takes the sunflower seedling to reach a height of 200 centimeters is approximately \(t = \frac{ln(\frac{56}{2600})}{-0.65}\) weeks. b. The asymptote of the function is \(h = 256\) cm, which represents the maximum height the sunflower seedling can reach in an unlimited amount of time.

Step-by-step solution

Set up the equation for the height

In the first part, to determine the time it takes for the sunflower to reach a height of 200 cm, set \(h(t)\) equal to 200 in the given logistic function: \(200 = \frac{256}{1+13 e^{-0.65 t}}\)

Solve for time

Rearranging this equation will eventually give \(t\). Begin by multiplying both sides by \(1+13e^{-0.65t}\), which gives: \(200 + 2600e^{-0.65t} = 256\). Then, subtract 200 from both sides: \(2600e^{-0.65t} = 56\). Afterwards, divide both sides by 2600 to isolate the exponential: \(e^{-0.65t} = \frac{56}{2600}\). To remove the exponent, take the natural logarithm of both sides: \(-0.65t = ln(\frac{56}{2600})\). Finally, solve for \(t\) by dividing both sides by -0.65: \(t = \frac{ln(\frac{56}{2600})}{-0.65}\)

Interpret the Asymptote

The asymptote of the logistic function represents the maximum height that the sunflower seedling can reach in a theoretically unlimited amount of time. In this case, the limit of the function \(h(t)\) as \(t\) approaches infinity is 256 cm, so the asymptote is \(h = 256\) on the graph.

Key concepts

Modeling Growth with the Logistic Function

The logistic function is a powerful tool used to model growth in various contexts, including biology, ecology, and social sciences. Unlike linear growth, which continues indefinitely, or exponential growth, which escalates rapidly without bounds, logistic growth starts rapidly and then slows down, approaching a maximum value called the 'carrying capacity'.

In the context of a sunflower seedling, the logistic function can reflect the more realistic scenario in which the seedling grows quickly at first and then its growth rate decreases as resources become scarce or the plant comes closer to its genetic potential height. The function given by
\[ h(t)=\frac{256}{1+13 e^{-0.65 t}} \]
describes this type of growth. The function encapsulates the dynamics of the seedling’s growth over time, where \(t\) represents the time in weeks, and \(h(t)\) represents the height of the seedling in centimeters. It’s important to understand how each parameter in this function influences the seedling's growth curve to make accurate predictions.

For instance, the 256 in the numerator represents the carrying capacity, or in this case, the maximum potential height of the sunflower. The growth rate is represented by 0.65, which determines how quickly the seedling approaches its carrying capacity. Finally, the value 13 affects how the growth starts. This modeling of growth using logistic functions is valuable in making predictions and in understanding how different factors may influence growth over time.

Solving Exponential Equations

One often encounters exponential equations when dealing with growth and decay problems. In the case of the sunflower seedling, to find the time \(t\) it takes to reach a certain height, we must solve an exponential equation derived from the logistic function. The initial step is always to isolate the exponential term, which can be achieved through operations that sequentially eliminate other terms, usually by performing inverse operations.

For example, to solve the equation
\[ 200 = \frac{256}{1+13 e^{-0.65 t}} \]
for our sunflower seedling's height of 200 centimeters, we engage in a series of algebraic maneuvers. Initially multiplying both sides by the denominator \(1+13e^{-0.65t}\) allows us to focus on the exponential term \(e^{-0.65t}\). After simplification, you should take the natural logarithm of both sides because the inverse of the exponential function is the natural logarithm, thus allowing us to solve for \(t\).

Understanding this process is beneficial, not just for one particular problem, but for a myriad of contexts where exponential relationships model the behavior of a system over time. Recognizing and correctly manipulating these equations is a fundamental skill in mathematics that plays a vital role in various scientific and engineering disciplines.

Interpretation of Asymptotes

In mathematics, an asymptote of a function is a line that the graph of the function approaches but never actually touches. Asymptotes can be horizontal, vertical, or diagonal, and each offers different insights regarding the behavior of the function.

For the given logistic growth function of a sunflower seedling, the horizontal asymptote is particularly significant. It illustrates a point where the growth stabilizes, indicating the carrying capacity of the system or the maximal value the variable of interest can attain under existing conditions. In our sunflower growth model, the asymptote is represented by
\[ h = 256 \] centimeters.

This implies that no matter how much time passes, the sunflower seedling will not exceed this height. Interpreting asymptotes in this context provides us with a clear understanding of long-term expectations. It helps us predict that after the seedling reaches a certain maturity, its growth will plateau, leading to a stable height. This is critical for gardeners, biologists, or anyone interested in the long-term growth patterns of plants or similar phenomena, as it sets realistic expectations and helps in planning for space, nutrients, and other resource allocations necessary for optimum growth.