Problem 74 - College Algebra and Calculus: An Applied Approach

Question

Part of the life cycle of a salmon is migration for reproduction. Salmon are anadromous fish. This means that they swim from the ocean to fresh water streams to lay their eggs. During migration, salmon must jump waterfalls to reach their destination. The path of a jumping salmon is given by \(h=-0.42 x^{2}+2.52 x\) where \(h\) is the height (in feet) and \(x\) is the horizontal distance (in feet) from where the salmon left the water. Will the salmon clear a waterfall that is 3 feet high if it leaves the water 4 feet from the waterfall?

Step-by-Step Solution

Answer

Yes, the salmon will clear the waterfall.

Step 1: Understand the given function and evaluate it at the right point

The height of the fish's jump is given by the function \(h=-0.42 x^{2} + 2.52 x\). To find out if the fish clears the waterfall, one needs to first evaluate this function at \(x = 4\), which is the horizontal distance from where the salmon left the water to the waterfall.

Step 2: Substitute \(x\) into the equation

Substitute \(x = 4\) into the equation \(h=-0.42 x^{2} + 2.52 x\) to get the height at this point.

Step 3: Solve the equation

Solving the equation gives \(h=-0.42 * 4^2 + 2.52 * 4 = -0.42 * 16 + 10.08 = -6.72 + 10.08 = 3.36\). So at \(x = 4\) the height is 3.36 ft.

Step 4: Compare the height with the waterfall's height

Now compare the height of the fish's jump with the height of the waterfall. Here, 3.36 ft is greater than 3 ft, so the salmon successfully clears the waterfall.

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions that can be represented in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The graph of a quadratic function is a parabola. This means it has a U-shape that either opens upward or downward.
Key characteristics of quadratic functions include:

• The vertex: The highest or lowest point on the parabola, depending on whether it opens downwards or upwards.
• The axis of symmetry: A vertical line that passes through the vertex and divides the parabola into two identical halves.
• The roots: The values of \( x \) where the quadratic intersects the \( x \)-axis. In some cases, these can be real or complex numbers.

In the context of our salmon jumping exercise, the function \( h = -0.42x^2 + 2.52x \) is a quadratic equation where the parabola opens downward because the leading coefficient (\(-0.42\)) is negative. This specific quadratic models the path of the salmon's jump, allowing us to calculate the maximum height reached during the jump.

Projectile Motion

Projectile motion describes the curved trajectory of an object thrown into the air under the influence of gravity. In nature, it applies to many scenarios, including the jump of a salmon. This kind of motion can be split into horizontal and vertical components.

• Horizontal motion is characterized by constant velocity, as there are no forces (ignoring air resistance) acting in this direction.
• Vertical motion is influenced by gravity, creating a uniformly accelerated motion.

The path of a projectile, like our salmon, is a parabola. The motion is typically analyzed using the principles of quadratic functions, where time or horizontal distance is the independent variable, and height is the dependent variable. This allows us to predict where and how high the salmon will jump at any given point.
In our exercise, the mathematical model doesn't account for time explicitly. Instead, it directly relates horizontal distance \( x \) to height \( h \), but the concepts of projectile motion help explain why the salmon follows a parabolic path.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world phenomena. This allows for predictions and analyses based on data. In our salmon example, mathematical modeling helps to forecast whether the fish will successfully clear the waterfall.
Key steps in mathematical modeling include:

• Defining the problem or scenario to be modeled. Here, it is the path of a jumping salmon.
• Establishing assumptions and selecting appropriate mathematical tools. We assume no air resistance and use a quadratic function to describe the trajectory.
• Applying and solving the mathematical tools to gather insights. We evaluated the function at \( x = 4 \) to determine the jump height.

By evaluating the formula \( h = -0.42x^2 + 2.52x \) at various points, and particularly at \( x = 4 \), we derive a clear, calculable method for determining if the salmon can clear the waterfall. This example shows how mathematical modeling can simplify complex natural behaviors into understandable quantities.