Problem 65 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The height (in feet) of a certain kind of tree is approximated by $$ h(t)=\frac{160}{1+240 e^{-0.2 t}} $$ where \(t\) is the age of the tree in years. Estimate the age of an 80 -ft tree.

Step-by-Step Solution

Answer

The age of an 80-feet tree can be calculated using the given height function \( h(t) = \frac{160}{1 + 240 e^{-0.2t}} \). After substituting h=80 and solving for t, we find that the age of the tree is approximately 16.09 years.

Step 1: Write down the given formula and information

The height function for the tree is given as: \( h(t) = \frac{160}{1 + 240 e^{-0.2t}} \) where t is the age in years, and h is the height in feet. We are given that the height, h = 80 feet.

Step 2: Substitute the height value into the formula and solve for t

Substitute h=80 into the height function: \( 80 = \frac{160}{1 + 240 e^{-0.2t}} \)

Step 3: Isolate the exponential term

To isolate the exponential term, multiply both sides of the equation by the denominator, and then divide by 80: \( 1 + 240e^{-0.2t} = \frac{160}{80} \) \( 1 + 240e^{-0.2t} = 2 \) Subtract 1 from both sides: \( 240e^{-0.2t} = 1 \) Divide by 240 to isolate the exponential term: \( e^{-0.2t} = \frac{1}{240} \)

Step 4: Take the natural logarithm of both sides of the equation

Take the natural logarithm (ln) of both sides of the equation to remove the exponential term: \( \ln{e^{-0.2t}} = \ln{\frac{1}{240}} \)

Step 5: Simplify and solve for t

Using the property of logarithms, we can simplify the left side of the equation: \( -0.2t = \ln{\frac{1}{240}} \) Divide by -0.2 to solve for t: \( t = \frac{\ln{\frac{1}{240}}}{-0.2} \) Calculate the value of t: \( t \approx 16.09 \) The age of an 80-feet tree is approximately 16.09 years.

Key Concepts

Applied Mathematics in Action

Applied mathematics involves using mathematical methods and models to solve real-world problems in various fields such as science, engineering, business, and more. It combines theoretical knowledge with practical applications. In our case, we've worked with an equation to determine the height of a tree at different ages, which is an example of applied mathematics within biology and environmental studies.

In this example, the formula provided is an application of exponential growth and decay, a common concept in applied mathematics. These types of equations are frequently used to model populations, radioactive decay, and in finance for calculating interest rates. By examining how the height of the tree changes over time according to the formula, we can gain insights into the tree's growth patterns and predict future growth. Understanding and being able to apply such mathematical concepts can be fundamental in managing resources and making informed decisions in the corresponding fields.

For students, applying mathematics in real-life situations not only helps to grasp the concepts more effectively but also encourages the development of problem-solving and analytical skills. Overall, applied mathematics helps to bridge the gap between abstract mathematical theories and their practical utility.

Deciphering Natural Logarithms

Natural logarithms, denoted as 'ln', are logarithms that have the base 'e', where 'e' is an irrational and transcendental number approximately equal to 2.71828. Natural logarithms are particularly useful when dealing with exponential growth and decay processes like the one in our tree height equation.

When you take the natural logarithm of both sides of an equation involving an exponential function with base 'e', such as in Step 4 of our solution, the logarithm function 'matches' the base of the exponential, effectively 'canceling' out the exponent and allowing us to solve for the variable within.

This logarithmic property is crucial because it simplifies our equation in a way that we can solve for the time variable, 't'. Thus, mastering natural logarithms is essential for students tackling exponential equations. It's a versatile tool not just in biology, but also in many areas like physics, finance, and information theory, where exponential relations are present.

Solving Exponential Equations

Solving exponential equations is a fundamental part of algebra and pre-calculus. These equations contain variables located in the exponent, which can often be solved using logarithms. The process generally involves isolating the exponential term and then taking the logarithm of both sides of the equation to bring down the exponent, just as we did in our tree height problem.

Our solution transformed an exponential equation into a linear one, making it possible to find the value of a variable. This process of isolating the variable is key to solving any algebraic equation. By understanding the steps of solving exponential equations, students can tackle a wide array of problems in science and finance, as exponential functions model a vast number of natural phenomena and financial trends.

In the context of education, it's crucial for students to learn the process of solving these equations in a methodical manner. Being able to solve exponential equations opens up a world of possibilities for investigating complex situations where variables grow or decay at rates proportional to their size, which is a common occurrence in the real world.