Problem 54 - Algebra and Trigonometry with Analytic Geometry

Question

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?

Step-by-Step Solution

Answer

(a) \(L = \frac{29}{7}t + 24\). (b) ~0.138 ft/day. (c) \(W = \frac{20}{7}t + 3\). (d) ~0.095 tons/day.

Step 1: Understanding the Linear Relationship - Length

Given two points for length, (0 months, 24 feet) and (7 months, 53 feet), we assume a linear relationship between length \(L\) and time \(t\). The general form of a linear equation is \(L = mt + c\), where \(m\) is the slope and \(c\) is the intercept. We can find \(m\) by calculating the change in length divided by the change in months: \(m = \frac{53 - 24}{7 - 0} = \frac{29}{7}\).

Step 2: Formulating the Linear Equation for Length

With \(m = \frac{29}{7}\) and using the point \((0, 24)\), the intercept \(c\) is 24. Thus, the equation for the length \(L\) as a function of age \(t\) is: \(L = \frac{29}{7}t + 24\).

Step 3: Calculating Daily Increase in Length

The slope \(\frac{29}{7}\) is the month-to-month increase, approximately 4.14 feet per month. To find the daily increase, divide by 30: \(\frac{4.14}{30} \approx 0.138\) feet per day.

Step 4: Understanding the Linear Relationship - Weight

Given two points for weight, (0 months, 3 tons) and (7 months, 23 tons), we assume a linear relationship between weight \(W\) and time \(t\). The slope \(m\) is calculated as: \(m = \frac{23 - 3}{7 - 0} = \frac{20}{7}\).

Step 5: Formulating the Linear Equation for Weight

With \(m = \frac{20}{7}\) and using the point \((0, 3)\), the intercept \(c\) is 3. Thus, the equation for the weight \(W\) as a function of age \(t\) is: \(W = \frac{20}{7}t + 3\).

Step 6: Calculating Daily Increase in Weight

The slope \(\frac{20}{7}\) is the month-to-month increase, approximately 2.86 tons per month. To find the daily increase, divide by 30: \(\frac{2.86}{30} \approx 0.095\) tons per day.

Key Concepts

Slope Calculation

Slope calculation is an essential concept in understanding linear relationships. It represents how much one variable, like length or weight, changes in response to another variable, usually time. To determine the slope, you calculate the difference in the values of the dependent variable, divided by the difference in the values of the independent variable. For instance, when looking at the increase in length of a whale over time, the slope is calculated as:

• Change in length / Change in time = (53 - 24) / (7 - 0) = 29 / 7

This gives us the change in length per month. Knowing how to calculate the slope helps us understand the rate at which changes occur. This method applies to any situation where two points determine a line, providing a straightforward way to measure the average rate of change.

Linear Equations

Linear equations are foundational in algebra and describe a straight line when plotted on a graph. The general form of a linear equation is given by:\[ y = mx + c \] Here, "\(m\)" represents the slope, and "\(c\)" is the y-intercept, the point where the line crosses the y-axis. To express the length \(L\) of the whale in terms of time \(t\), we substitute our slope and initial length into the equation to get:

• \(L = \frac{29}{7}t + 24\)

Similarly, for weight \(W\):

• \(W = \frac{20}{7}t + 3\)

Understanding linear equations allows you to model relationships where one quantity depends linearly on another.

Unit Conversion

Unit conversion is a practical skill in problem solving, particularly in converting measurements to ensure consistency throughout calculations. In this exercise, converting months to days facilitates finding daily changes. For instance, the monthly increase in whale length is 4.14 feet. To find the daily increase, we divide by the number of days in a month (30):

• \(\frac{4.14}{30} \approx 0.138\) feet per day

Similarly, converting the weight change from tons per month to tons per day involves:

• \(\frac{2.86}{30} \approx 0.095\) tons per day

Accurate unit conversions are crucial for understanding the correct scale and pace of changes in measurements.

Problem Solving

Problem solving in mathematics often involves breaking down complex problems into simpler parts. In this exercise, students are tasked with finding the relationships between time and two variables: length and weight. Here are some useful strategies:

• Identify what is given and what needs to be found.
• Understand the relationships and express them through linear equations.
• Convert units where necessary to simplify calculations and ensure accuracy.

An excellent approach when tackling problems is to steadily progress through each step, ensuring answers are logically derived from the posed equations and calculations. By practicing these skills, students enhance their ability to model real-world situations mathematically, leading to a deeper comprehension of the subject matter.