Problem 22 - Applied Calculus

Question

Allometry is the study of the relative size of different parts of a body as a consequence of growth. In this problem, you will check the accuracy of an allometric equation: the weight of a fish is proportional to the cube of its length. \(^{87}\) Table 1.40 relates the weight, \(y,\) in \(\mathrm{gm}\), of plaice (a type of fish) to its length, \(x,\) in \(\mathrm{cm} .\) Does this data support the hypothesis that (approximately) \(y=k x^{3} ?\) If so, estimate the constant of proportionality, \(k\) $$\begin{array}{r|r|r|r|r|r} \hline x & y & x & y & x & y \\ \hline 33.5 & 332 & 37.5 & 455 & 41.5 & 623 \\ 34.5 & 363 & 38.5 & 500 & 42.5 & 674 \\ 35.5 & 391 & 39.5 & 538 & 43.5 & 724 \\ 36.5 & 419 & 40.5 & 574 & & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The data supports the hypothesis; estimated \( k \approx 0.00876 \).

Step 1: Understand the equation y = kx^3

We need to test if the relationship between the weight and length of the fish is approximately represented by the equation \( y = kx^3 \), where \( k \) is a constant of proportionality. This implies that the weight \( y \) of the fish should be proportional to the cube of its length \( x \).

Step 2: Calculate x^3 for each x value

Compute the cube of each length \( x \) given in the table: \( x = 33.5, 34.5, 35.5, ... , 43.5 \).- \( 33.5^3 = 37517.375 \)- \( 34.5^3 = 41093.625 \)- \( 35.5^3 = 44692.375 \)- \( 36.5^3 = 48402.625 \)- \( 37.5^3 = 52734.375 \)- \( 38.5^3 = 57096.625 \)- \( 39.5^3 = 61509.375 \)- \( 40.5^3 = 65972.625 \)- \( 41.5^3 = 71527.375 \)- \( 42.5^3 = 76568.125 \)- \( 43.5^3 = 82318.875 \)

Step 3: Calculate k for each pair of (x^3, y) values

For each pair of \( (x^3, y) \), calculate \( k \) using the equation \( k = \frac{y}{x^3} \).- For \( x = 33.5, k = \frac{332}{37517.375} \approx 0.00884 \)- For \( x = 34.5, k = \frac{363}{41093.625} \approx 0.00883 \)- For \( x = 35.5, k = \frac{391}{44692.375} \approx 0.00875 \)- For \( x = 36.5, k = \frac{419}{48402.625} \approx 0.00866 \)- For \( x = 37.5, k = \frac{455}{52734.375} \approx 0.00863 \)- For \( x = 38.5, k = \frac{500}{57096.625} \approx 0.00876 \)- For \( x = 39.5, k = \frac{538}{61509.375} \approx 0.00875 \)- For \( x = 40.5, k = \frac{574}{65972.625} \approx 0.00870 \)- For \( x = 41.5, k = \frac{623}{71527.375} \approx 0.00871 \)- For \( x = 42.5, k = \frac{674}{76568.125} \approx 0.00880 \)- For \( x = 43.5, k = \frac{724}{82318.875} \approx 0.00880 \)

Step 4: Analyze the values of k

Looking at the calculated values of \( k \), they are all approximately in the range of 0.00863 to 0.00884. This consistent pattern suggests that \( y \) is indeed proportional to \( x^3 \).

Step 5: Estimate the constant of proportionality k

Estimate \( k \) by averaging the individual \( k \) values obtained. The average is: \[ k \approx \frac{0.00884 + 0.00883 + 0.00875 + 0.00866 + 0.00863 + 0.00876 + 0.00875 + 0.00870 + 0.00871 + 0.00880 + 0.00880}{11} \approx 0.00876 \]

Key Concepts

Proportional Relationships

Understanding proportional relationships is essential in many fields, including biology and physics. In a proportional relationship, two quantities increase or decrease together at the same rate. This means that if one quantity doubles, the other also doubles, maintaining a constant ratio between them.
For instance, in the context of the allometry exercise, we hypothesize that the weight of a fish (\( y \)) is proportional to the cube of its length (\( x^3 \)). This relationship is expressed mathematically as \( y = kx^3 \). Here \( k \) is a constant – a fixed number that specifies the exact proportionality relationship for these variables.
To determine if the hypothesis holds true, we must check if the ratio \( \frac{y}{x^3} \) remains approximately constant across various lengths and weights.

• A consistent ratio suggests a proportional relationship.
• An inconsistent ratio indicates other factors might be at play.

Proportional relationships are visually represented by a straight line if you plot one quantity against the other, but since we deal with \( x^3 \) instead of \( x \), the plot involves nonlinear scaling.

Allometric Equation

The allometric equation is a specific form of a power-law relationship used to describe how characteristics of living organisms change with size. It can be expressed in the form \( y = kx^a \), where \( k \) and \( a \) are constants. In biology, these relationships often describe how one biological variable, such as mass, varies with another, like length.
The exercise focuses on the hypothesis that fish weight is proportional to the cube of its length, implying a power of three. Therefore, the allometric equation simplifies to \( y = kx^3 \), where \( y \) is the fish's weight, and \( x \) is its length.
To apply this equation:

• Calculate the cube of the fish's length for each instance in the data.
• Check each calculated weight against these values using the equation.

If the relationship holds, it means biological growth processes within the fish species maintain this specific proportional pattern. Allometric equations help scientists understand growth dynamics and can be applied to many other contexts in biology.

Constant of Proportionality

The constant of proportionality is a crucial component in proportional relationships. It indicates how much one quantity changes in relation to another. In our allometric equation \( y = kx^3 \), \( k \) represents this constant and is central to defining the relationship between the fish's length and weight.
In this problem, to find \( k \), divide the weight \( y \) by the cube of the length \( x^3 \) for each data point. This gives different \( k \) values for each pair of measurements.

• Compute \( k = \frac{y}{x^3} \)
• Examine the variation across calculated values
• A minimal variation indicates a strong consistent proportionality

A consistent value of \( k \) across different sets of data would confirm the allometric hypothesis. By averaging these values, you get an estimate for \( k \), allowing predictions of fish weight for other lengths.