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Chapter 2: Problem 54

Growth of a blae whate Newborn blue whales are approximately 24 feet long and weigh 3 tons. Young whales are nursed for 7 months, and by the time of weaning they often are 53 feet long and weigh 23 tons. Let \(L\) and \(W\) denote the length (in feet) and the weight (in tons), respectively, of a whale that is \(t\) months of age. (a) If \(L\) and \(t\) are linearly related, express \(L\) in terms of \(t\) (b) What is the daily increase in the length of a young whale? (Use 1 month \(=30\) days.) (c) If \(W\) and \(t\) are linearly related, express \(W\) in terms of \(t\) (d) What is the daily increase in the weight of a young whale?

Short Answer

Expert verified
(a) \(L(t) = 4.14t + 24\); (b) 0.138 feet/day; (c) \(W(t) = 2.86t + 3\); (d) 0.095 tons/day.

Step by step solution

01

Define the Linear Relationship for Length

We start by defining the linear relationship between length \(L\) and time \(t\). We know that the function follows the form \(L(t) = mt + b\), where \(m\) is the rate of increase per month, and \(b\) is the initial length at \(t = 0\). Given: \(L(0) = 24\) feet and \(L(7) = 53\) feet.
02

Calculate the Rate of Change for Length

Using the given lengths at \(t = 0\) and \(t = 7\), the rate of change \(m\) is calculated as \(m = \frac{L(7) - L(0)}{7 - 0} = \frac{53 - 24}{7 - 0} = 4.14\) feet per month.
03

Formulate the Linear Equation for Length

From Step 1, the equation is \(L(t) = mt + b\). Substituting \(m = 4.14\) and \(b = 24\), we have \(L(t) = 4.14t + 24\).
04

Calculate Daily Increase in Length

The daily increase in length is the monthly rate of change divided by 30 days: \(\frac{4.14}{30} = 0.138\) feet per day.
05

Define the Linear Relationship for Weight

Similarly, define the linear relationship between weight \(W\) and time \(t\) using \(W(t) = nt + c\). Given: \(W(0) = 3\) tons and \(W(7) = 23\) tons.
06

Calculate the Rate of Change for Weight

Calculate \(n\) as \(n = \frac{W(7) - W(0)}{7 - 0} = \frac{23 - 3}{7 - 0} = 2.86\) tons per month.
07

Formulate the Linear Equation for Weight

The equation is \(W(t) = nt + c\). Using \(n = 2.86\) and \(c = 3\), we have \(W(t) = 2.86t + 3\).
08

Calculate Daily Increase in Weight

The daily increase in weight is the monthly rate of change divided by 30 days: \(\frac{2.86}{30} = 0.095\) tons per day.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are mathematical expressions that describe a straight line on a graph. They are given in the form of \( y = mx + b \), where \( m \) is the slope or rate of change, and \( b \) is the y-intercept, representing the starting value when \( t = 0 \).
This form is commonly used to describe relationships where changes occur at a constant rate over time, such as the growth of a blue whale in size.
In this exercise, we used the linear equation \( L(t) = 4.14t + 24 \) to represent the length \( L \) of a whale as a function of age \( t \) in months.
Rate of Change
The rate of change tells us how much one quantity changes in relation to another. It is calculated as the difference in the output divided by the difference in the input.
For this scenario, the rate of change is crucial for understanding how quickly the whale grows in length and weight over time.
  • For length, we calculated a monthly rate of 4.14 feet per month.
  • For weight, the rate was determined to be 2.86 tons per month.
These rates show the constant progression of growth, highlighting the whale鈥檚 development pattern.
Biological Modeling
Biological modeling uses mathematical structures to represent and analyze real-world biological processes. It helps us comprehend the dynamics of living organisms, like the growth of a blue whale.
By employing linear models for length and weight, we can effectively predict these attributes based on the whale's age.
This specific model assumes steady growth rates, simplifying complex biological processes into a manageable framework that students can understand and analyze.
Mathematical Modeling
Mathematical modeling involves creating equations or systems of equations representing real-life phenomena. In our exercise, we utilized mathematical equations to model the growth of a blue whale over time.
Mathematical models, such as those using linear equations, provide us with a coherent way to predict outcomes and make adjustments or predictions related to real-life events.
  • For length, the model \( L(t) = 4.14t + 24 \) helps us calculate how long the whale will be at any month.
  • For weight, \( W(t) = 2.86t + 3 \) allows computation of the expected weight.
These models are powerful tools for connecting mathematical concepts with biological understanding.

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Most popular questions from this chapter

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