Question

MODELING WITH MATHEMATICS The length \(L\) (in millimeters) of the larvae of the black porgy fish can be modeled by $$ L(x)=0.00170 x^2+0.145 x+2.35,0 \leq x \leq 40 $$ where \(x\) is the age (in days) of the larvae. Write and solve an inequality to find at what ages a larva's length tends to be greater than 10 millimeters. Explain how the given domain affects the solution.

Short answer

The larva's length tends to be greater than 10 millimeters when the larva is approximately 24 days old.

Step-by-step solution

Understand the Problem

The problem requires finding the age at which the length of the larvae tends to be greater than 10 mm. This requires solving the inequality \(L(x) > 10\). Given the function \(L(x)=0.00170 x^2+0.145 x+2.35\), we need to replace \(L(x)\) with 10 and solve for \(x\).

Set up the Inequality

Replace \(L(x)\) with 10 in the given function: \(0.00170 x^2+0.145 x+2.35 > 10\).

Simplify the Inequality

To simplify, subtract 10 from both sides giving us \(0.00170 x^2+0.145 x-7.65 > 0\).

Solve the Inequality

To solve the quadratic inequality, we use the quadratic formula giving \(x = \frac{-0.145±\sqrt{((0.145)^2-4*0.00170*(-7.65))}}{2*0.00170} \) . After calculating, we get \(x \approx 23.98\) and \(x \approx 179.98\).

Apply the Domain

However, the domain provided is 0 ≤ \(x\) ≤ 40, meaning the \(x\) value must lie within this range. Hence \(x \approx 23.98\) is the only valid solution.

Interpret the Results

Therefore, the larvae’s length will be greater than 10 mm when it’s approximately 24 days. Applying the domain, we can see that the larvae could only reach 10mm within its first 40 days of life.

Key concepts

Modeling with mathematics

When we talk about modeling with mathematics, we refer to using mathematical tools to represent real-world situations. In our problem, we're modeling the growth of black porgy fish larvae using a quadratic function.

This mathematical model, expressed as \(L(x)=0.00170 x^2+0.145 x+2.35\), helps us understand how the larvae's length changes with time. By turning the growth process into a mathematical equation, we can predict and analyze different growth stages.

Mathematical modeling is crucial as it lets us explore, simulate, and understand complex systems without physically testing every scenario. In this context, it's particularly useful for biologists who study fish growth patterns. They can use the function to hypothesize about growth potential over time without invasive procedures.

Quadratic functions

Quadratic functions are a type of polynomial function with the form \(ax^2 + bx + c\). They are common in mathematical modeling due to their ability to describe parabolic shapes. In our fish larvae problem, the quadratic function \(L(x)=0.00170 x^2+0.145 x+2.35\) shows how length grows as time progresses.

Key properties of quadratic functions include:

• Their graphs form parabolas, which are symmetric curves.
• The vertex of the parabola represents either the maximum or minimum point of the function.
• Depending on the leading coefficient \(a\) being positive or negative, the parabola opens upwards or downwards, respectively.

Solving quadratic inequalities involves finding the range of \(x\) for which the function produces desired outputs, like lengths greater than 10 mm. This often includes using the quadratic formula and analyzing the vertex and intercepts.

Domain and range

The concepts of domain and range are essential in understanding the limitations and potentials of a function, especially in real-world applications. The domain refers to all possible input values \(x\) that a function can accept. Likewise, the range is all possible output values \(L(x)\).

For the fish larvae problem, the domain \(0 \leq x \leq 40\) indicates we're only interested in larvae's ages from 0 to 40 days. This constraint is essential since, outside these values, the model might be invalid or biologically irrelevant.

The range in our context concerns finding the age when the length becomes greater than 10 mm. By combining both domain and range, we effectively narrow down possible solutions from mathematical computations to realistic, meaningful scenarios. This benefits researchers and scientists in making informed decisions based on accurate data.