Problem 31 - Algebra 2

Question

BOATING. For Exercises 30 and \(31,\) use the following information. A motor boat traveling against waves accelerates from a resting position. Suppose the speed of the boat in feet per second is given by the function \(f(t)=-0.04 t^{4}+0.8 t^{3}+0.5 t^{2}-t,\) where \(t\) is the time in seconds. It takes 6 seconds for the boat to travel between two buoys while it is accelerating. Use synthetic substitution to find \(f(6)\) and explain what this means.

Step-by-Step Solution

Answer

\( f(6) = 22.16 \); speed at 6 seconds is 22.16 ft/s.

Step 1: Identify the Function and Given Time

We are provided with the speed function of the boat as \( f(t) = -0.04 t^{4} + 0.8 t^{3} + 0.5 t^{2} - t \). We need to find \( f(6) \) using synthetic substitution, meaning we substitute \( t = 6 \) into the function.

Step 2: Set Up Synthetic Substitution

Synthetic substitution involves using a simplified method similar to synthetic division to evaluate a polynomial. In this setup, we list the coefficients of the polynomial, which are: \(-0.04, 0.8, 0.5, -1\). We align them for substitution.

Step 3: Initialize the Synthetic Substitution Process

Start the synthetic substitution process by placing \(6\) (the given \( t \) value) on the left side, write the leading coefficient \((-0.04)\) below the line, and bring it down directly for use in the next steps.

Step 4: Multiply and Add Iteratively

- Multiply \(-0.04\) by \(6\), then add the result to the next coefficient (\(0.8\)).- Repeat this by using each resulting value, multiply it by \(6\), and add it to the next coefficient step-by-step: - \(-0.04 \times 6 = -0.24\), add \(0.8\), result is \(0.56\). - \(0.56 \times 6 = 3.36\), add \(0.5\), result is \(3.86\). - \(3.86 \times 6 = 23.16\), add \(-1\), result is \(22.16\).

Step 5: Conclude the Calculation and Interpretation

The final result of this synthetic substitution is \( f(6) = 22.16 \), meaning the speed of the boat at \( t = 6 \) seconds is \( 22.16 \) feet per second.

Key Concepts

Polynomial Function Evaluation

Polynomial function evaluation involves finding the output of a polynomial function by plugging in a specific value for the variable. In this exercise, we have a polynomial function that represents the speed of a boat: \( f(t) = -0.04t^4 + 0.8t^3 + 0.5t^2 - t \). To evaluate this at \( t = 6 \), we need to substitute \( t \) with 6, which allows us to calculate the boat's speed at this time.

Synthetic substitution is a creative mathematical technique used to streamline the process of evaluating such polynomials. Instead of plugging \( t = 6 \) directly into each term, we use a procedure akin to synthetic division, which simplifies the work and reduces potential calculation errors. This 'synthetic' approach helps in quickly assessing the function's value at any specific input, hence facilitating efficient polynomial function evaluation for tasks like speed determination.

Speed Calculation

Speed calculation in this context refers to determining how fast the motorboat is traveling at a certain time during its acceleration. For our problem, the polynomial function \( f(t) = -0.04t^4 + 0.8t^3 + 0.5t^2 - t \) represents speed in terms of feet per second based on the variable \( t \), which is time in seconds.

We specifically want to know the speed of the boat at 6 seconds. By performing synthetic substitution, we found that \( f(6) = 22.16 \) feet per second. This result signifies that, after accounting for all acceleration effects and resistance from waves, the boat achieves a speed of 22.16 feet per second at this precise moment.

Algebraic Problem-Solving

Algebraic problem-solving involves using algebraic techniques to solve for unknown variables or evaluate expressions. This exercise is a classic demonstration of applying algebraic principles to assess a real-world situation, the motion of a motorboat.

Through the use of synthetic substitution, which optimizes traditional polynomial evaluation methods, we efficiently solve for the speed of the boat in mere seconds. Algebraic problem-solving not only aids in simplifying complex calculations but also in making sense of relationships defined by algebraic expressions. This is particularly helpful in systematically analyzing and interpreting polynomial mathematical models.

Function Analysis

Function analysis involves scrutinizing a function to comprehend its behavior fully across different values. In this particular exercise, the given function \( f(t) \), which models boat speed, needs careful evaluation to understand how the speed changes over time as \( t \) varies.

Analyzing a function like this one helps reveal patterns in acceleration and how various factors contribute to the changing speed of the boat. By knowing the speed at designated times, one can, for example, make predictions or optimize settings for better performance. Function analysis becomes an invaluable tool in interpreting complex phenomena and consistently applying mathematical concepts to real-life scenarios.