Problem 84 - Intermediate Algebra

Question

Oceanography. The width \(w\) (in millimeters) of successive growth spirals of the sea shell Catapulus voluto, shown below, is given by the exponential function \(w(n)=1.54 e^{0.503 n}\) where \(n\) is the spiral number. Find the width, to the nearest tenth of a millimeter, of the sixth spiral.

Step-by-Step Solution

Answer

The width of the sixth spiral is approximately 31.5 mm.

Step 1: Identify the given formula

The problem provides us with the exponential function that describes the width of the spirals. The function is given as \( w(n) = 1.54 e^{0.503 n} \), where \( n \) is the spiral number.

Step 2: Substitute the spiral number

Since we are asked to find the width of the sixth spiral, substitute \( n = 6 \) into the given formula. This gives us \( w(6) = 1.54 e^{0.503 \, \times \, 6} \).

Step 3: Simplify the exponent

Calculate the exponent part of the expression: \( 0.503 \, \times \, 6 = 3.018 \). Therefore, the expression simplifies to \( w(6) = 1.54 e^{3.018} \).

Step 4: Calculate the exponential value

Use a calculator to find \( e^{3.018} \). This gives approximately \( 20.481 \).

Step 5: Calculate the width of the sixth spiral

Now multiply the calculated exponential value by 1.54: \( w(6) = 1.54 \, \times \, 20.481 \). This gives approximately \( 31.54 \).

Step 6: Round the result to the nearest tenth

Round 31.54 to the nearest tenth to obtain the final answer. This results in \( 31.5 \) millimeters.

Key Concepts

Growth Spirals

Growth spirals, found in nature, are fascinating patterns that showcase how living organisms grow over time. In seashells, these spirals often display an exponential pattern, where each growth segment or spiral is a bit larger than the previous. This natural design not only exhibits beauty but also efficiency. For example, the seashell of the Catapulus voluto displays such spiraling growth. The width of each spiral increases exponentially, resulting in more significant growth with each successive spiral.

Understanding growth spirals involves recognizing the pattern they follow. It is not linear; instead, it follows an exponential path, where each new spiral builds upon the previous one using a predictable mathematical growth function. This function helps us determine how much larger each subsequent spiral will be, using constants derived from biological growth patterns. Growth spirals provide insight into natural growth processes, allowing mathematicians and scientists to predict and analyze patterns found in nature.

Calculating Width

When calculating the width of growth spirals, as seen in the Catapulus voluto seashell, we rely on mathematical functions, particularly exponential ones. An exponential function uses a base number raised to a power to give us precise measurements. In this case, the function for the seashell spirals is \( w(n) = 1.54 e^{0.503 n}\) where \( e \) is the mathematical constant approximately equal to 2.718.

To find the width of a specific spiral, you substitute the spiral number into the equation. For example, to find the width of the sixth spiral, replace \( n \) with 6 in the formula. This is the same as evaluating the function mathematically by first calculating the exponent and then the entire expression.

• Identify the required spiral number.
• Substitute into the function.
• Use your calculator to find the result.

This process gives an accurate measure of each spiral's growth, providing a deeper understanding of how each section of the seashell expands.

Exponential Growth

Exponential growth is a type of growth where the rate becomes ever faster by a constant factor over equal increments of time or order. Unlike linear growth, which adds a constant amount in each step, exponential growth multiplies by a constant factor. In the seashell example, this means that each growth spiral increases in width based on a consistent exponential factor.

The function \( w(n) = 1.54 e^{0.503 n}\) represents exponential growth since each subsequent value of \( w(n) \) grows faster than the previous values.

• Exponential growth starts slowly but accelerates rapidly.
• It's represented mathematically through a base and exponent, often using \( e \).
• This type of growth pattern is widespread in natural phenomena.

Exponential growth models like this help illustrate how small differences in time or generation can lead to substantial increases in growth and size, which is vital for understanding patterns in biology, population dynamics, and even finance.