Question

The surface area \(A\) (in square meters) of a person with a mass of 60 kilograms can be approximated by \(A=0.2195 h^{0.3964}\), where \(h\) is the height (in centimeters) of the person. a. Find the inverse function. Then estimate the height of a 60-kilogram person who has a body surface area of \(1.6\) square meters. b. Verify that function \(A\) and the inverse model in part (a) are inverse functions.

Short answer

The inverse function is \(h= (0.2195 A)^{1/0.3964}\). The estimated height of a 60-kilogram person who has a body surface area of \(1.6\) square meters is the result of the equation \(h= (0.2195 * 1.6)^{1/0.3964}\). Verification shows that the two functions are indeed inverses.

Step-by-step solution

Finding the Inverse Function

The first step in finding the inverse function is to switch the dependent variable \(A\) with the independent variable \(h\). So, instead of \(A=0.2195 h^{0.3964}\), we get \(h = 0.2195 A^{0.3964}\). Next, isolate \(h\) to find the inverse function. This is done by raising both sides of the equation to the power of \(1/0.3964\). That is: \(h= (0.2195 A)^{1/0.3964}\).

Estimating the Height

To estimate the height of a person with a body surface area of \(1.6\) square meters, simply plug the value of \(A\) into the inverse function. In other words: \(h= (0.2195 * 1.6)^{1/0.3964}\). Solve this to get the approximate height in centimeters.

Verifying the Inverse Function

Finally, verify the inverse function by showing that \(A(h(h^{-1}(A))) = A\) for any number \(A\) and \(A(h^{-1}(h(A))) = h(A)\) for any number \(h\). Since the output of the inverse function is the same as the original output of the regular function, we can confirm that they are inverse functions.

Key concepts

Surface Area Approximation

Surface area approximation is an interesting application in mathematics and biology that helps to estimate the surface area of a human body based on simple measurements like height and mass. This is commonly used in medicine for dosing medications or assessing medical risks. In this case, we have a formula that is used to approximate the surface area of a 60-kilogram person given their height in centimeters, represented as \(A=0.2195 h^{0.3964}\). This function allows for quick estimates without needing complex tools or equations.
When dealing with surface area approximation, keep in mind that it's an estimate. It's not 100% accurate, but it's useful in contexts where precision is less critical. Approximating with an established formula like this helps make quick decisions based on just a couple of measurements. Always remember that in scientific models, such approximations are a gateway to understanding but should be used with consideration of their inherent simplifications.

Inverse Function Verification

Inverse function verification is the process of proving that two functions are inverses of each other. This means essentially that you can "undo" one function by applying the other. For the surface area model, we found an inverse function by switching the roles of height and surface area in the equation. This gave us \(h = (0.2195 A)^{1/0.3964}\).
Verifying an inverse involves checking two conditions:

• Applying one function after its inverse should return the initial input: \(A(h(h^{-1}(A))) = A\).
• Applying the inverse after the function should also return the initial input: \(h(A(h^{-1}(h(A)))) = h\).

These verifications show that once you input them into each other, they produce the starting number, proving they are indeed inverse functions. This is crucial for cases where where deriving the inverse is non-trivial or where errors in derivation might occur.

Mathematical Modeling

Mathematical modeling involves creating equations and functions to represent real-world problems. This allows us to use mathematics to solve and understand phenomena in various fields, from science to engineering. In our problem, the equation \(A=0.2195 h^{0.3964}\) models how the human body surface area relates to height.
A good model should be:

• Accurate enough to reliably solve the problem it was designed for.
• Simplified so that it can be easily used and understood by someone with basic math skills.
• Flexible so that it can be adapted to slightly different situations.

Models function as a bridge from real-world contexts to mathematical abstraction and back again. They provide predictions, explanations, and a framework for testing quantitative hypotheses. This exercise’s model is an easy-to-use tool that simplifies the surface area calculation and provides insight into the nature of the relationship between height and surface area.