Question

the surface area of a mammal, s, satisfies the equation ( s = km^{\frac{2}{3}} ), where m is the body mass, and the constant of proportionality k depends on the body shape of the mammal. a human of body mass 70 kilograms has surface area 18,600 ( \text{cm}^2 ). find the constant of proportionality for humans. find the surface area of a human with body mass 59 kilograms. round your answers to the nearest integer. ( k = ) 1095 the surface area of a human with body mass 59 kilograms is ( \text{cm}^2 ).

Explanation:

Step1: Find the constant \( k \)

We know the formula \( S = kM^{\frac{2}{3}} \), and for a human with \( M = 70 \) kg, \( S = 18600 \) \( \text{cm}^2 \). Substitute these values into the formula to solve for \( k \):
\[
18600 = k \times 70^{\frac{2}{3}}
\]
First, calculate \( 70^{\frac{2}{3}} \). We can rewrite it as \( (70^{\frac{1}{3}})^2 \). \( 70^{\frac{1}{3}} \approx 4.121 \), so \( (4.121)^2 \approx 17.0 \) (more accurately, using a calculator: \( 70^{\frac{2}{3}}=\sqrt[3]{70^2}=\sqrt[3]{4900}\approx 16.98 \)). Then:
\[
k=\frac{18600}{70^{\frac{2}{3}}}\approx\frac{18600}{16.98}\approx 1095
\]

Step2: Calculate the surface area for \( M = 59 \) kg

Now that we have \( k \approx 1095 \), use the formula \( S = kM^{\frac{2}{3}} \) with \( M = 59 \):
First, calculate \( 59^{\frac{2}{3}} \). \( 59^{\frac{1}{3}}\approx 3.893 \), so \( (3.893)^2\approx 15.16 \) (more accurately, \( 59^{\frac{2}{3}}=\sqrt[3]{59^2}=\sqrt[3]{3481}\approx 15.16 \)). Then:
\[
S = 1095\times 15.16
\]
\[
S\approx 1095\times 15.16 = 1095\times(15 + 0.16)=1095\times 15+1095\times 0.16 = 16425+175.2 = 16600.2
\]
Rounding to the nearest integer, \( S\approx 16600 \).

Answer:

The surface area of a human with body mass 59 kilograms is approximately \( \boxed{16600} \) \( \text{cm}^2 \).