Question

The recommended percent \(p\) of nitrogen (by volume) in the air that a diver breathes is given by \(p=\frac{105.07}{d+33}\), where \(d\) is the depth (in feet) of the diver. Find the depth when the air contains \(47 \%\) recommended nitrogen by (a) solving an equation, and (b) using the inverse of the function.

Short answer

In both solutions (a) by solving an equation and (b) using the inverse function, the resulting depth \(d\) found is roughly 0.238 ft.

Step-by-step solution

Write down the equation with given value

First, write down the given equation \(p=\frac{105.07}{d+33}\) and insert the given value for \(p\) which is 47. Hence the equation to solve becomes \(47=\frac{105.07}{d+33}\).

Solve the equation for \(d\)

Now, rearrange the equation to solve for \(d\). Start by multiplying both sides by \(d+33\) to get \(47(d+33)=105.07\). Then, expanding the brackets and solving for \(d\) gives \(d = \frac{105.07}{47} - 33\).

Calculate \(d\)

Now you can easily calculate \(d\) using a calculator. Here, \(d\) is roughly 0.238 ft.

Write down the inverse function

The inverse function of \(p=\frac{105.07}{d+33}\) is obtained by swapping \(d\) and \(p\), then rearranging the resulting equation to make \(d\) the subject. Here, the inverse function is \(d=\frac{105.07}{p}-33\)

Find \(d\) using the inverse function

Substitute \(p=47\) into the inverse function to find \(d\). That means \(d=\frac{105.07}{47}-33\). Use the calculator to get the value of \(d\). The depth \(d\) here is also roughly 0.238 ft.

Key concepts

Inverse Functions

An inverse function reverses the effect of a function. If a function \( f \) takes an input \( x \) and gives an output \( y \), then its inverse, often written as \( f^{-1} \), takes \( y \) as an input and returns \( x \). In other words, the inverse function undoes what the original function did.
For our exercise, the given function is \( p = \frac{105.07}{d+33} \). Here, we treat \( p \) as the output and \( d \) as the input. By performing algebraic manipulation, we form the inverse, which allows us to calculate \( d \) when \( p \) is known.
To find the inverse, you swap \( p \) and \( d \), resulting in \( d = \frac{105.07}{p} - 33 \). By substituting \( p = 47 \), you can directly calculate the specific value of \( d \) for \( 47\% \) nitrogen.

Solving Equations

Solving equations is all about finding the value of an unknown variable that makes the equation true. We start with the equation \( 47 = \frac{105.07}{d+33} \) and need to solve for \( d \).
To isolate \( d \), multiply both sides by \( d + 33 \). This clears the fraction, leading to \( 47(d+33) = 105.07 \). Distribute the \( 47 \) across \( (d + 33) \), giving \( 47d + 1551 = 105.07 \).
Subtract 1551 from both sides to continue solving for \( d \), resulting in \( 47d = 105.07 - 1551 \). Divide both sides by \( 47 \) to find \( d \approx \frac{105.07}{47} - 33 \). Completing these steps with a calculator verifies the calculated depth \( d \) is approximately 0.238 ft. This approach shows a systematic method for equation solving, useful in many math problems.

Rational Functions

Rational functions consist of ratios of polynomials. In our exercise’s function \( p = \frac{105.07}{d+33} \), you can see it as a rational function where the numerator is constant, and the denominator is a first-degree polynomial.
This function models a relationship that varies based on \( d \), the depth. As \( d \) increases, the denominator grows, and hence the overall value of \( p \) decreases. This inverse variation is a characteristic trait of rational functions.
Understanding rational functions is crucial as they model real-life situations where one quantity depends on the inverse of another, like our diving example. Here, it signifies the inverse relationship between the depth dive and the percentage of nitrogen needed, showing practical applications beyond pure computation.