Question

The length \(\ell\) (in centimeters) of a scalloped hammerhead shark can be modeled by the function $$ \ell=266-219 e^{-0.05 t} $$ where \(t\) is the age (in years) of the shark. How old is a shark that is 175 centimeters long?

Short answer

Therefore, the age of the shark is approximately equal to 15 years when rounded to the nearest whole number.

Step-by-step solution

Substitute Length into Equation

Substitute 175 for \(\ell\) in the equation, getting the equation to be solved for \(t\): 175 = 266 - 219 e^{-0.05 t}

Isolate the Exponential Term

Isolate the exponential term on one side of the equation by subtracting 266 from both sides: -91 = -219 e^{-0.05 t}

Simplify the Equation

Simplify the equation by dividing it by -219 on both sides: 0.416 = e^{-0.05 t}

Apply the Natural Logarithm

Taking natural logarithm on both sides to clear the base e: ln(0.416) = -0.05 t

Solve for t

Finally, solve for \(t\) by dividing both sides by -0.05: \(t = \frac{ln(0.416)}{-0.05}\)

Key concepts

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world phenomena using mathematical concepts and language. By creating a mathematical model, we can analyze, predict, or explain the behavior of different elements within a given context. In the original exercise, the growth of the scalloped hammerhead shark is represented using an exponential function, which is a common technique in biology for modeling population growth, radioactive decay, and similar processes.

In this case, the length \(\ell\) of a shark is modeled as a function of its age \(t\). The model \(\ell=266-219 e^{-0.05 t}\) incorporates an exponential decay factor to reflect the rate at which the growth slows down as the shark ages. Mathematics provides us with the tools to describe this natural pattern in a precise, quantifiable way, which then allows scientists and researchers to make predictions about the shark's size at any given age.

Natural Logarithms

Natural logarithms are a fundamental concept in mathematics, particularly when dealing with exponential growth and decay. The natural logarithm, often represented as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828182846.

This particular constant arises naturally in many areas of mathematics, such as compound interest, calculus, and, of course, the growth and decay models we see in biology.

The power of the natural logarithm comes from its ability to reverse the effect of the exponential function. When you take the natural logarithm of an exponential function like \(e^{x}\), what you're left with is simply \(x\), the exponent. This property is exactly what's used in the problem's step-by-step solution; by applying the natural logarithm to both sides of the equation, we are able to isolate the variable \(t\) and solve for it.

Solving Exponential Equations

When faced with an exponential equation, the goal is to isolate and solve for the unknown variable, which in our case is the age of the shark, \(t\). Exponential equations have variables in the exponent and can seem daunting at first. However, by performing a series of algebraic manipulations and using logarithms, we can find solutions to these equations.

As illustrated by the provided steps, solving an exponential equation typically involves isolating the term with the exponent and then using natural logarithms to eliminate the base \(e\), making the variable accessible. The steps in the exercise moved from isolating the exponential term to taking the natural logarithm of both sides to linearize the equation — this made it possible to solve for time \(t\).

It's important to understand these steps because exponential equations appear not just in theoretical exercises, but in real-world applications across sciences, finance, and engineering where they're used to model growth, decay, and many types of change over time.