Problem 30 - 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. Find the speed of the boat at \(1,2,\) and 3 seconds.

Step-by-Step Solution

Answer

The speeds are 0.26 ft/s at 1s, 5.76 ft/s at 2s, and 19.86 ft/s at 3s.

Step 1: Understand the Function

The given function is \( f(t) = -0.04t^4 + 0.8t^3 + 0.5t^2 - t \), which describes the speed of the boat at any time \( t \) in seconds. The goal is to evaluate this function to find the speed at specific time intervals: 1, 2, and 3 seconds.

Step 2: Substitute t = 1 into the Function

To find the speed at \( t = 1 \), substitute 1 into the function: \( f(1) = -0.04(1)^4 + 0.8(1)^3 + 0.5(1)^2 - 1(1) \). Simplifying this, we have: \( f(1) = -0.04 + 0.8 + 0.5 - 1 = 0.26 \text{ ft/s} \).

Step 3: Substitute t = 2 into the Function

To find the speed at \( t = 2 \), substitute 2 into the function: \( f(2) = -0.04(2)^4 + 0.8(2)^3 + 0.5(2)^2 - 2(1) \). Simplifying this, we calculate: \( f(2) = -0.64 + 6.4 + 2 - 2 = 5.76 \text{ ft/s} \).

Step 4: Substitute t = 3 into the Function

To find the speed at \( t = 3 \), substitute 3 into the function: \( f(3) = -0.04(3)^4 + 0.8(3)^3 + 0.5(3)^2 - 3(1) \). Simplifying this, we find: \( f(3) = -3.24 + 21.6 + 4.5 - 3 = 19.86 \text{ ft/s} \).

Key Concepts

Substitution

Substitution is a powerful technique commonly used in algebra to find the value of a variable within a given function. It involves replacing the variable with a specific number, which allows us to calculate the function's output at that particular point.In the case of the speed of the boat, given by the function \( f(t) = -0.04t^4 + 0.8t^3 + 0.5t^2 - t \), we need to substitute the time \( t \) with specific values to understand how quickly the boat is moving at those times.

Here's how substitution works:

• Identify the variable: Determine which variable needs to be replaced. In this problem, it is \( t \) (time in seconds).
• Choose specific values: Decide the values you want to evaluate. Here, it's \( t = 1, 2, \text{and } 3 \).
• Replace the variable: Substitute each value individually into the function and simplify each equation to find the result.

This approach helps you find specific outputs of the function, revealing how conditions change over time.

Function Evaluation

Function evaluation refers to the process of calculating the output of a function for a particular input value. It helps in determining the result of applying the function with the given input after substitution.

When evaluating the polynomial function \( f(t) = -0.04t^4 + 0.8t^3 + 0.5t^2 - t \), it becomes crucial for understanding the speed of the boat at various times. Here's how you can evaluate a function:

• Substitute the input value: For a specific \( t \), substitute it in place throughout the function, as seen in this process for \( t = 1, 2, \text{and } 3 \).
• Simplify the expression: Break down each term step by step, ensuring arithmetic operations are done correctly. This includes raising numbers to powers, multiplying coefficients, and adding or subtracting terms.
• Verify the calculation: Double-check the arithmetic to prevent errors, especially in polynomial functions where multiple operations are involved.

This method allows you to understand changes in outputs, such as a boat's speed, depending on time or other factors input into functions.

Speed Calculation

Speed calculation in physics and its related fields often involves analyzing a function that models motion, speed, or velocity. Given the polynomial function \( f(t) = -0.04t^4 + 0.8t^3 + 0.5t^2 - t \), evaluating it gives us the boat's speed at specific time intervals.

It's essential to understand a few fundamental speed calculation concepts:

• Units Matter: In this context, speed is measured in feet per second (ft/s), a standard unit of speed.
• Time Intervals: Examine how speed changes with time by substituting different time values (in this case \( t = 1, 2, ext{and } 3 \)).
• Real-life Relevance: Knowing the boat's speed at certain moments helps in practical navigation and understanding of boat performance over time.

By calculating the speed at different times, we learn about the boat's acceleration patterns and how external factors like the waves affect its motion. This understanding is crucial for both theoretical and practical applications in fields like boating, physics, and engineering.