Problem 123
Question
Height and Weight The formula \(W=\frac{25}{7} h-\frac{800}{7}\) approximates the recommended minimum weight for a person \(h\) inches tall, where \(62 \leq h \leq 76.\) (a) What is the recommended minimum weight for someone 70 inches tall? (b) Does \(W\) represent a one-to-one function? (c) Find a formula for the inverse. (d) Evaluate the inverse for 150 pounds and interpret the result. (e) What does the inverse compute?
Step-by-Step Solution
Verified Answer
(a) 135.71 pounds; (b) Yes; (c) \(h = \frac{7W + 800}{25}\); (d) 74 inches; (e) Height for a given weight.
1Step 1: Understanding the Formula
The given formula is \( W = \frac{25}{7} h - \frac{800}{7} \), which calculates the recommended minimum weight \( W \) for a person of height \( h \) inches. The height \( h \) must be between 62 and 76.
2Step 2: Calculate Recommended Weight for 70 Inches
To find the recommended minimum weight for someone with a height of 70 inches, substitute \( h = 70 \) into the formula: \[ W = \frac{25}{7} \times 70 - \frac{800}{7} = \frac{1750}{7} - \frac{800}{7} = \frac{950}{7} = 135.71 \] The weight is approximately 135.71 pounds.
3Step 3: Understanding One-to-One Function
A function is one-to-one if each input maps to a unique output. Here, the function \( W = \frac{25}{7} h - \frac{800}{7} \) is linear with a non-zero slope, indicating it is one-to-one.
4Step 4: Finding the Inverse Function
To find the inverse, solve for \( h \) in terms of \( W \). Start from the equation: \[ W = \frac{25}{7} h - \frac{800}{7} \] Multiply through by 7 to clear the fractions: \[ 7W = 25h - 800 \] Add 800 to both sides: \[ 25h = 7W + 800 \] Divide by 25: \[ h = \frac{7W + 800}{25} \] This is the inverse function.
5Step 5: Evaluating the Inverse for 150 Pounds
Using the inverse formula \( h = \frac{7W + 800}{25} \), substitute \( W = 150 \): \[ h = \frac{7 \times 150 + 800}{25} = \frac{1050 + 800}{25} = \frac{1850}{25} = 74 \] The result is a height of 74 inches.
6Step 6: Interpreting the Inverse Computation
The inverse computes the height corresponding to a given weight, reversing the original process. For instance, a 150-pound person would have a recommended height of 74 inches according to the inverse.
Key Concepts
Inverse Functions in AlgebraUnderstanding Linear EquationsThe Role of Mathematical ModelingExploring Function Properties
Inverse Functions in Algebra
Inverse functions are fascinating parts of algebraic functions. An inverse function essentially reverses the action of the original function. If you imagine putting something through the original function and receiving an output, the inverse function takes that output and gives you back the original input.
To find an inverse function, you have to solve the equation to express the original variable in terms of the new variable. For example, in the formula for the recommended minimum weight, we switched from expressing weight in terms of height to expressing height in terms of weight.
This process is like working backward through the problem-solving steps of the original function. It's crucial to note that not every function has an inverse (it must be one-to-one).
To find an inverse function, you have to solve the equation to express the original variable in terms of the new variable. For example, in the formula for the recommended minimum weight, we switched from expressing weight in terms of height to expressing height in terms of weight.
This process is like working backward through the problem-solving steps of the original function. It's crucial to note that not every function has an inverse (it must be one-to-one).
Understanding Linear Equations
Linear equations are the backbone of many mathematical models, including the one in this problem. They represent straight lines when plotted on a graph. A linear equation has the general form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
The function given in the exercise \(W = \frac{25}{7} h - \frac{800}{7}\) is a linear equation. This indicates a constant rate of change, represented by the slope (\(\frac{25}{7}\)). The y-intercept (-\(\frac{800}{7}\)) tells you where the line crosses the y-axis. Understanding this helps to grasp how weight changes as height changes.
With linear equations, every change in height results in a proportional change in weight, making them predictable and easy to use for modeling.
The function given in the exercise \(W = \frac{25}{7} h - \frac{800}{7}\) is a linear equation. This indicates a constant rate of change, represented by the slope (\(\frac{25}{7}\)). The y-intercept (-\(\frac{800}{7}\)) tells you where the line crosses the y-axis. Understanding this helps to grasp how weight changes as height changes.
With linear equations, every change in height results in a proportional change in weight, making them predictable and easy to use for modeling.
The Role of Mathematical Modeling
Mathematical modeling is the process of representing real-world scenarios with mathematical expressions. In this exercise, the formula given is a mathematical model. It approximates how height relates to weight using a linear equation.
Models help us make informed predictions and better understand real-world situations. They simplify complexities by using mathematics. However, it's important to remember that models are approximations. They are only as good as the assumptions and data they're based on.
In our situation, the model provides a guideline for the minimum recommended weight based on height. While it can't capture every unique individual's characteristics, it offers a general rule that can be very useful for health and fitness considerations.
Models help us make informed predictions and better understand real-world situations. They simplify complexities by using mathematics. However, it's important to remember that models are approximations. They are only as good as the assumptions and data they're based on.
In our situation, the model provides a guideline for the minimum recommended weight based on height. While it can't capture every unique individual's characteristics, it offers a general rule that can be very useful for health and fitness considerations.
Exploring Function Properties
Understanding function properties is key in determining how functions behave. A function's properties can include aspects like its domain, range, whether it's one-to-one, and if it has an inverse.
For the function \(W = \frac{25}{7} h - \frac{800}{7}\), we know it's a one-to-one function. This means each height has one unique weight. It passes the horizontal line test, meaning any horizontal line drawn through its graph will only cross it once.
This unique relationship is why we were able to find an inverse function. Functions with non-overlapping outputs are prime candidates for finding inverses, making them versatile in various mathematical applications.
For the function \(W = \frac{25}{7} h - \frac{800}{7}\), we know it's a one-to-one function. This means each height has one unique weight. It passes the horizontal line test, meaning any horizontal line drawn through its graph will only cross it once.
This unique relationship is why we were able to find an inverse function. Functions with non-overlapping outputs are prime candidates for finding inverses, making them versatile in various mathematical applications.
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