Problem 26 - Applied 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 0.855 m². (b) Weight is approximately 84.2 kg. (c) Height function: \( h = \left(\frac{s}{0.02888}\right)^{\frac{4}{3}} \).

Step 1: Substitute values into the formula

For part (a), we need to find the surface area for a person who weighs 65 kg and is 160 cm tall. We substitute the values of weight \(w = 65\) kg and height \(h = 160\) cm into the formula \(s = 0.01 w^{0.25} h^{0.75}\). This gives us: \[s = 0.01 \times 65^{0.25} \times 160^{0.75}\].

Step 2: Calculate components in the formula

Calculate \(65^{0.25}\) and \(160^{0.75}\). First, compute \(65^{0.25} = 2.718\). Next, compute \(160^{0.75} = 31.496\).

Step 3: Find the surface area

Now multiply the results from the previous step and the constant: \(s = 0.01 \times 2.718 \times 31.496 = 0.855\). Therefore, the surface area of the person in part (a) is approximately \(0.855\, \mathrm{m}^2\).

Step 4: Rearrange formula to solve for weight

For part (b), to find the weight \(w\) of a person with given height \(h = 180\, \mathrm{cm}\) and surface area \(s = 1.5\, \mathrm{m}^2\), rearrange the formula: \[w^{0.25} = \frac{s}{0.01 \cdot h^{0.75}}\].

Step 5: Substitute known values

Substitute \(s = 1.5\, \mathrm{m}^2\) and \(h = 180\, \mathrm{cm}\) into the rearranged equation: \[w^{0.25} = \frac{1.5}{0.01 \times 180^{0.75}} \approx \frac{1.5}{0.01 \times 49.482} = 3.031\].

Step 6: Solve for weight

Now solve for \(w\) by raising \(3.031\) to the fourth power: \(w = 3.031^4 \approx 84.2\, \mathrm{kg}\). So the weight is approximately 84.2 kg.

Step 7: Rearrange equation for fixed weight

For part (c), we are given a fixed weight \(w = 70\, \mathrm{kg}\) and need to solve for \(h\) as a function of \(s\). Substitute \(w = 70\) into the original formula: \[s = 0.01 \times 70^{0.25} \times h^{0.75}\]. Rearrange to solve for \(h^{0.75}\): \(h^{0.75} = \frac{s}{0.01 \times 70^{0.25}}\).

Step 8: Simplify the expression for height

Calculate \(70^{0.25} = 2.888\), then substitute into the equation \(h^{0.75} = \frac{s}{0.01 \times 2.888}\). Simplifying, \(h^{0.75} = \frac{s}{0.02888}\). Finally, solve for \(h\) by raising both sides to the \(\frac{4}{3}\) power: \(h = \left(\frac{s}{0.02888}\right)^{\frac{4}{3}}\).

Key Concepts

Surface Area Calculation

Calculating the surface area of a human body can be important in many fields, such as medicine and physiology. The DuBois formula is a reliable method for approximating this surface area. It uses a person's weight and height in the calculation, reflecting the influence of both dimensions on body surface area. For example, to find the surface area of a person who weighs 65 kg and is 160 cm tall, the formula requires substituting these values into:

• \( s = 0.01 imes 65^{0.25} imes 160^{0.75} \)

By calculating each component separately and then combining them, we can determine the surface area measurement, an important factor in determining how much of certain medications a person should receive.

Weight and Height Relation

The DuBois formula highlights a significant relationship between weight and height in determining a body's surface area. When solving for a person's weight, given their height and surface area, the formula is rearranged accordingly. For instance, if we know a person has a surface area of 1.5 \( ext{m}^2\) and height of 180 cm:

• The formula can be adjusted to solve for weight:
• \( w^{0.25} = \frac{1.5}{0.01 imes 180^{0.75}} \)

After computing, you can find the weight by solving for \( w \). This illustrates how both height and surface area contribute to weight-dependent characteristics.

Mathematical Modeling

Mathematical modeling helps us understand real-world phenomena, like body surface area, through equations. In the case of the DuBois formula, it exemplifies how math can be used to create a usable model that captures the relationship between physical attributes. By using powers and root calculations, the formula mathematically mimics the biological principles involved, offering a way to predict or estimate physiological metrics using basic statistical relationships. Understanding how each parameter

• - weight
• - height
• - calculated constants

functions within the equation is critical for its application in various scientific investigations and health assessments.

Applied Calculus Problem

The DuBois formula serves as an applied calculus problem where students can exercise their skills in manipulating equations and understanding functions. Part (c) of the exercise introduces solving for height as a function of surface area by fixing the weight at 70 kg. This involves rearranging complex power equations into simpler forms:

• \( h^{0.75} = \frac{s}{0.01 imes 70^{0.25}} \)
• Then calculating:
• \( h = \left(\frac{s}{0.02888}\right)^{\frac{4}{3}} \)

Through this exercise, students employ calculus concepts to find solutions, reinforcing their understanding of both calculus operations and real-world applications of these techniques. Such problems enhance critical thinking and problem-solving skills, essential for advanced mathematical studies.