Problem 84
Question
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
Verified Answer
The width of the sixth spiral is 31.6 mm.
1Step 1: Understand the Problem
We need to find the width of the sixth spiral in the sea shell's growth pattern using the given exponential function. The function provided is \( w(n) = 1.54 e^{0.503 n} \), where \( n \) represents the spiral number. In this case, \( n = 6 \).
2Step 2: Substitute the Spiral Number
Substitute the spiral number \( n = 6 \) into the function \( w(n) = 1.54 e^{0.503 n} \). This gives us \( w(6) = 1.54 e^{0.503 imes 6} \).
3Step 3: Calculate the Exponent
Calculate the exponent \( 0.503 imes 6 \). This gives \( 3.018 \). Substitute this value back into the equation so it becomes \( w(6) = 1.54 e^{3.018} \).
4Step 4: Evaluate the Exponential Expression
Use a calculator to evaluate \( e^{3.018} \). This is approximately \( 20.491 \). Substitute this back into the expression to get \( w(6) = 1.54 imes 20.491 \).
5Step 5: Multiply and Round to Nearest Tenth
Multiply \( 1.54 \) by \( 20.491 \) to get \( 31.556 \). Rounding this to the nearest tenth gives \( 31.6 \).
Key Concepts
Exponential GrowthMathematical ModelingAlgebraic Expressions
Exponential Growth
Exponential growth is a process where quantities increase at a constant percentage rate over time. In our exercise, the spirals of a sea shell grow exponentially, defined by the function \( w(n) = 1.54 e^{0.503 n} \). Exponential functions involve exponents and the base of natural logarithms, \( e \), which is approximately equal to \( 2.718 \).
When graphed, exponential growth appears as a curve that rises steeply after a certain point, illustrating how rapidly the quantity can increase. This is evident in the shell's spiral expansion, where each subsequent spiral becomes significantly larger. As \( n \) increases, \( e^{0.503 n} \) grows rapidly, leading to a much larger \( w(n) \) value.
Understanding exponential growth is crucial in fields such as biology, economics, and environmental science, as it helps model systems like population growth, compound interest, and more.
When graphed, exponential growth appears as a curve that rises steeply after a certain point, illustrating how rapidly the quantity can increase. This is evident in the shell's spiral expansion, where each subsequent spiral becomes significantly larger. As \( n \) increases, \( e^{0.503 n} \) grows rapidly, leading to a much larger \( w(n) \) value.
Understanding exponential growth is crucial in fields such as biology, economics, and environmental science, as it helps model systems like population growth, compound interest, and more.
Mathematical Modeling
Mathematical modeling involves the use of mathematical expressions and equations to represent real-world phenomena. The aim is to create a simplified version of reality, allowing us to predict outcomes and understand complex systems better.
In this exercise, the growth of the sea shell is modeled using the equation \( w(n) = 1.54 e^{0.503 n} \). This model uses the concept of exponential functions to describe how each spiral in the shell grows in width. Through the model, we can predict the width of any spiral by substituting in the spiral number \( n \).
Mathematical models rely on assumptions to keep them simple and useful. While they cannot capture every detail of reality, models like the one used here provide valuable insights into patterns and behaviors of natural phenomena.
In this exercise, the growth of the sea shell is modeled using the equation \( w(n) = 1.54 e^{0.503 n} \). This model uses the concept of exponential functions to describe how each spiral in the shell grows in width. Through the model, we can predict the width of any spiral by substituting in the spiral number \( n \).
Mathematical models rely on assumptions to keep them simple and useful. While they cannot capture every detail of reality, models like the one used here provide valuable insights into patterns and behaviors of natural phenomena.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In our problem, the expression \( w(n) = 1.54 e^{0.503 n} \) represents the width of the spiral as an ongoing function of \( n \), the spiral number.
An algebraic expression allows us to perform calculations to find specific outcomes, like the width of the sixth spiral. By substituting \( n = 6 \), we transform the general expression into a specific equation that we can solve with arithmetic and functions of \( e \).
Using algebraic expressions, we frame real-world problems into mathematical forms that are easier to work with. This makes solving complex puzzles, like predicting the size of natural patterns, more manageable. Remember, substituting values and ensuring correct calculation pathways is essential for accuracy.
An algebraic expression allows us to perform calculations to find specific outcomes, like the width of the sixth spiral. By substituting \( n = 6 \), we transform the general expression into a specific equation that we can solve with arithmetic and functions of \( e \).
Using algebraic expressions, we frame real-world problems into mathematical forms that are easier to work with. This makes solving complex puzzles, like predicting the size of natural patterns, more manageable. Remember, substituting values and ensuring correct calculation pathways is essential for accuracy.
Other exercises in this chapter
Problem 84
If \(f(x)=x^{2}+1\) and \(g(x)=3 x^{2}-2,\) show that \((f \circ g)(x) \neq(g \circ f)(x)\)
View solution Problem 84
Write logarithmic expression as one logarithm. \(\ln \left(x y+y^{2}\right)-\ln (x z+y z)+\ln z\)
View solution Problem 85
Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ e^{3 x}=9 $$
View solution Problem 85
Write logarithmic expression as one logarithm. \(\frac{1}{2} \log _{6}\left(x^{2}+1\right)-\log _{6}\left(x^{2}+2\right)\)
View solution