Problem 34 - Prealgebra

Question

MARINE BIOLOGY Manatees can weigh up to 1000 pounds and are generally no more than 10 feet long. Suppose a manatee is currently 6.25 feet long. Write and solve an inequality to find how much longer the manatee could grow.

Step-by-Step Solution

Answer

The manatee can grow an additional 3.75 feet at most.

Step 1: Define the Length Limit

According to the problem, a manatee can be up to 10 feet long. This is the maximum length a manatee can reach.

Step 2: Set Up the Inequality

Let the variable \( x \) represent the additional length that the manatee can grow. Since the manatee is currently 6.25 feet long, the inequality is: \( 6.25 + x \leq 10 \). This inequality states that the current length plus the additional length \( x \) cannot exceed 10 feet.

Step 3: Solve the Inequality

Subtract 6.25 from both sides of the inequality to isolate \( x \): \[ x \leq 10 - 6.25 \]Calculate the right side: \[ 10 - 6.25 = 3.75 \]Thus, the solution to the inequality is: \[ x \leq 3.75 \].

Step 4: Interpret the Solution

The solution \( x \leq 3.75 \) means that the manatee can grow an additional 3.75 feet at most to remain within the 10 feet length limit.

Key Concepts

Understanding Inequalities

Inequalities are like equations, but instead of showing that two sides are equal, they show that one side is larger or smaller than the other.
They are useful in situations where you need to express limits or boundaries, such as determining the maximum length a manatee can grow.
In an inequality, the symbols used are:

• 5 (less than)
• 5= (less than or equal to)
• > (greater than)
• >= (greater than or equal to)

These help us set boundaries in real-world problems. For our manatee, we wrote the inequality \(6.25 + x \leq 10\) to say that the current length plus possible growth can't exceed 10 feet.
This lets us solve for how much further the manatee can grow.

Exploring Problem Solving

Problem-solving in mathematics often means breaking down a problem into smaller, manageable steps.
In the manatee example, we began by understanding the scenario: the manatee's current and maximum lengths.
The next step was to translate this understanding into mathematical terms by setting up an inequality. Solving this inequality involved basic arithmetic steps, like subtraction.
Finally, we interpreted what the solution meant in practical terms, communicating how much more the manatee could grow.
Remember, the key steps are:

• Understand the problem context.
• Set up a mathematical representation.
• Solve using arithmetic operations.
• Interpret the results in the context of the problem.

Introduction to Marine Biology

Marine biology is the study of marine organisms, their behaviors, and interactions with the environment. Manatees are a fascinating subject in marine biology.
They are large, gentle mammals who live in coastal and freshwater environments.
Understanding aspects like growth limits is crucial for their conservation, since it helps scientists assess the health and development of these species.
As shown in our example problem, knowing the maximum length aid biologists in monitoring their wellbeing. It influences habitat requirements and our understanding of their ecological niche.

Length Measurement in Mathematics

Length measurement is an essential skill in math, and it often appears in many real-world contexts. For example, the exercise shows a manatee with a current length of 6.25 feet.
Length allows us to apply mathematical concepts like inequalities and equations to determine possible future scenarios.
In practical situations, being able to measure and calculate lengths is fundamental. It helps us compare, estimate, and solve problems involving space and size.
Whether it’s wildlife, architecture, or everyday life, mastering length measurement is key in making informed decisions. Using this skill in combination with inequalities can solve questions about size limits, as seen in the manatee growth problem.