Problem 31 - Calculus

Question

The DuBois formula relates a person's surface area \(s\) in \(\mathrm{m}^{2},\) to weight \(w,\) in \(\mathrm{kg},\) and height \(h,\) in \(\mathrm{cm},\) by $$s=0.01 w^{0.25} h^{0.75}$$ (a) What is the surface area of a person who weighs \(65 \mathrm{kg}\) and is \(160 \mathrm{cm}\) tall? (b) What is the weight of a person whose height is \(180 \mathrm{cm}\) and who has a surface area of \(1.5 \mathrm{m}^{2} ?\) (c) For people of fixed weight \(70 \mathrm{kg}\), solve for \(h\) as a function of \(s .\) Simplify your answer.

Step-by-Step Solution

Answer

(a) Surface area is approximately 1.96 m². (b) Weight is approximately 9.32 kg. (c) The height is \(h = \left(\frac{s}{0.02788}\right)^{\frac{4}{3}}\)."

Step 1: Apply DuBois Formula for Surface Area

To find the surface area, apply the DuBois formula for a person with weight \(w = 65\) kg and height \(h = 160\) cm. Calculate \[s = 0.01 \times (65)^{0.25} \times (160)^{0.75}\]

Step 2: Calculate Surface Area using Formula

First, calculate \(65^{0.25}\) and \(160^{0.75}\).\[65^{0.25} \approx 2.748data160^{0.75} \approx 71.501\]Now substitute these values back into the formula:\[s = 0.01 \times 2.748 \times 71.501 \approx 1.964 \, \text{m}^2\]Thus, the surface area is approximately \(1.96 \, \text{m}^2\).

Step 3: Determine Weight given Surface Area and Height

For a person with height \(h = 180\) cm and surface area \(s = 1.5 \text{ m}^2\), solve for weight \(w\) using the formula:\[1.5 = 0.01 \times w^{0.25} \times 180^{0.75}\]Calculate \(180^{0.75} \approx 85.937\). Now solve:\[1.5 = 0.01 \times w^{0.25} \times 85.937\]\(w^{0.25} = \frac{1.5}{0.01 \times 85.937} \approx 1.746\).Raise both sides to the power of 4 to find \(w\):\[w = (1.746)^4 \approx 9.32 \]The weight is approximately \(9.32 \) kg.

Step 4: Solve for Height with Fixed Weight

For a fixed weight \(w = 70\) kg, express height \(h\) as a function of surface area \(s\). Start from:\[s = 0.01 \times 70^{0.25} \times h^{0.75}\]Calculate \(70^{0.25} \approx 2.788\), thus:\[s = 0.01 \times 2.788 \times h^{0.75}\]Rearrange to solve for \(h^{0.75}\):\[h^{0.75} = \frac{s}{0.01 \times 2.788}\]Raise both sides to the power of \(\frac{4}{3}\) to solve for \(h\):\[h = \left(\frac{s}{0.02788}\right)^{\frac{4}{3}}\]This gives the function for height \(h\) in terms of surface area \(s\).

Key Concepts

Surface Area Calculation

The DuBois formula is a mathematical equation used to calculate the surface area of a person based on their weight and height. The formula is expressed as:

• \(s = 0.01 w^{0.25} h^{0.75}\)

The surface area, denoted by \(s\), is measured in square meters (\(m^2\)), weight \(w\) is in kilograms, and height \(h\) is in centimeters.
This formula helps in assessing the surface area for medical, nutritional, or physiological assessments, ensuring easy and accurate calculation based on measurable parameters.
To find the surface area of someone weighing 65 kg and 160 cm tall, you use the formula by substituting these values in:

• Calculate \(65^{0.25}\) which gives approximately 2.748
• Calculate \(160^{0.75}\) which gives approximately 71.501

Substituting these back into the original equation gives:

• \(s = 0.01 \times 2.748 \times 71.501 \approx 1.964\, m^2\)

This means the approximate surface area is 1.96 square meters.

Weight-Height Relationship

The relationship between weight and height plays a crucial role in determining body surface area. The DuBois formula also allows us to solve for other variables, like weight, when given surface area and height.
Consider a scenario where a person has a height of 180 cm and a surface area of 1.5 \(m^2\). To find their weight \(w\), you substitute and rearrange the formula.

• The equation becomes \(1.5 = 0.01 \cdot w^{0.25} \cdot 180^{0.75}\)
• Calculate \(180^{0.75}\) which results in approximately 85.937

Rearranging gives:

• \(w^{0.25} = \frac{1.5}{0.01 \times 85.937} \approx 1.746\)

Solving for \(w\), you raise both sides to the fourth power:

• \(w = (1.746)^4 \approx 9.32\)

This solution finds the weight to be approximately 9.32 kg, given the height and surface area.

Solving Algebraic Equations

In situations where you have a fixed value for one variable, such as weight, while needing to express another variable like height in terms of surface area, algebraic manipulation comes into play.
Suppose the weight of a person is fixed at 70 kg, and we need to find height as a function of surface area \(s\). Start with the equation:

• \(s = 0.01 \cdot 70^{0.25} \cdot h^{0.75}\)

Calculate \(70^{0.25}\) which gives about 2.788, transforming the equation to:

• \(s = 0.01 \times 2.788 \times h^{0.75}\)

Rearranging the formula to solve for \(h^{0.75}\):

• \(h^{0.75} = \frac{s}{0.02788}\)

To solve for \(h\), raise both sides to the power of \(\frac{4}{3}\):

• \(h = \left(\frac{s}{0.02788}\right)^{\frac{4}{3}}\)

This technique highlights how algebraic rearrangement can make complex calculations more straightforward by expressing unknowns in terms of known quantities.