Question

PROBLEM SOLVING The wind speed \(s\) (in miles per hour) near the center of a tornado can be modeled by \(s=93 \log d+65\), where \(d\) is the distance (in miles) that the tornado travels. a. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the center of the tornado. b. Find the inverse of the given function. Describe what the inverse represents.

Short answer

a. The wind speed was approximately 282 miles per hour.\nb. The inverse function is \(s = 10^{(d - 65)/93}\), and it tells us how far a tornado would travel given its wind speed.

Step-by-step solution

Compute Wind Speed

To calculate the wind speed near the center of the 1925 tornado, substitute \(d = 220\) into the provided expression for \(s\).\nThat is, \(s = 93 \log (220) + 65\).

Solve for Wind Speed

Use your calculator's logarithm function to evaluate \( \log(220)\) and then substitute the result into the equation to solve for \(s\). Show your work: \(s = 93 \log (220) + 65 = 93 \times 2.342 + 65 = 282.2\). The wind speed near the center of the tornado would be approximately 282 miles per hour.

Inverse Function

Next, determine the inverse of the function \(s = 93 \log (d) + 65\). For such cases, begin by exchanging \(s\) and \(d\), to get \(d = 93 \log (s) + 65\), then solve for \(s\).

Solve for \(d\)

First, isolate logarithm on one side: \(d - 65 = 93 \log (s)\). The equation becomes \(\log(s) = (d - 65)/93\). Use properties of logarithm to find \(s\). Finally, we get the inverse function: \(s = 10^{(d - 65)/93}\).

Interpret Inverse Function

The inverse function gives distance as a function of speed, hence it predicts how far a tornado would travel given its wind speed.

Key concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions that involve logarithms. Logarithms are the inverse operations of exponential functions, meaning they help us find the exponent when the base and the result are known.

When we talk about a logarithmic function like \( f(x) = \log_b(x) \), it essentially asks: "To what power must we raise \( b \) to obtain \( x \)?" For example, in our exercise, the function \( s = 93 \log(d) + 65 \) uses the common logarithm log base 10 (often simply written as \( \log \)).

• Here, the variable \( d \) represents the input (distance), and \( s \) gives us the output (wind speed).
• Understanding logarithms is essential in solving equations where the unknown appears as the exponent of some variable.
• They are particularly useful in representing phenomena that grow or decay exponentially, such as natural disasters and sound intensity.

Tornado Math Problems

Tornado math problems often involve the application of mathematical models to represent natural phenomena, allowing us to make predictions or solve practical questions about these events.

In this exercise, we have a model for calculating wind speed near the center of a tornado based on the distance it travels. Such problems require understanding the relationships between different factors, like how increased distance can affect wind speed.

• Tornado models often utilize logarithmic equations because they effectively handle large ranges of data and provide a more realistic scale for understanding changes.
• Using the provided equation, \( s = 93 \log(d) + 65 \), we estimate values by substituting real data, such as in the historical example of the 1925 tornado.
• By substituting \( d = 220 \), the formula provides an estimation of wind speed near the tornado’s center.

Such problems help improve comprehension of how mathematical equations can be applied to real-life scenarios, making seemingly abstract concepts more tangible.

Wind Speed Calculations

Wind speed calculations provide valuable estimations based on mathematical models. They help predict the intensity of a tornado given certain data, such as the distance traveled.

In our specific example, the formula \( s = 93 \log(d) + 65 \) is utilized to find the wind speed. This formula considers the logarithm of the distance \( d \), emphasizing how even a small increase in distance can significantly affect wind speed calculations.

• By calculating \( \log(220) \) and proceeding with the multiplication and addition steps, we achieve an estimated wind speed of approximately 282 mph.
• This systematic approach allows one to efficiently predict critical weather conditions without needing to collect wind speed data directly—a process that might be dangerous during tornado conditions.
• Mathematically modeling wind speeds through equations like these can provide reliable estimations that aid in disaster preparedness and response planning.

Problem Solving in Algebra

Problem solving in algebra often involves finding unknowns and interpreting relationships through equations. In our exercise, not only did we compute values, but we also explored inverse functions.

Inverse functions in algebra essentially "undo" the action of the original function, allowing students to uncover alternative perspectives of the same relationship.

• To find an inverse, such as switching \(d\) and \(s\) in \(s = 93 \log(d) + 65\), we solve it for the new dependent variable.
• Through rearranging and isolating logarithmic terms, we derived the inverse function: \(s = 10^{(d - 65)/93}\).
• This function demonstrates how far a tornado would travel at a given wind speed, shifting the focus from wind speed to distance.

The exercise illuminates problem-solving methods that involve turning an original problem inside-out to gain new insights and applications. If a student can master these techniques, they can tackle similar algebraic challenges with confidence and creativity.