Question

The equation \(h=0.019 s^2\) models the height \(h\) (in feet) of the largest ocean waves when the wind speed is \(s\) knots. Compare the wind speeds required to generate 5 -foot waves and 20 -foot waves.

Short answer

The wind speeds required to generate 5-foot waves and 20-foot waves are approximately 16.22 knots and 32.45 knots respectively.

Step-by-step solution

Plug in the first height into the wave height equation

First, replace \(h\) in the equation \(h=0.019s^2\) with 5 to get \(5=0.019s^2\).

Solve the equation for wind speed s

Next, isolate \(s\) by dividing both sides by 0.019. This gives \(s^2 \approx 263.16\). Then, take the square root of both sides (since the wind speed cannot be negative) to get \(s \approx 16.22\) knots.

Plug in the second height into the wave height equation

Now replace \(h\) in the equation \(h=0.019s^2\) with 20 to get \(20=0.019s^2\).

Solve this new equation for wind speed s

Again, isolate \(s\) by dividing by 0.019, yielding \(s^2 \approx 1052.63\). Then, take the square root of both sides (since the wind speed cannot be negative) to get \(s \approx 32.45\) knots.

Key concepts

Solving Quadratic Equations

Grasping the method to solve quadratic equations is crucial in algebra, as these equations appear frequently in various mathematical scenarios. A quadratic equation is generally represented by the formula \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are coefficients and \(a \eq 0\). To solve it, there are several methods, such as factoring, completing the square, or using the quadratic formula.

In the case of the ocean wave height equation \(h = 0.019s^2\), we are presented with a simpler form where the equation is already set to equal \(h\), we can consider it a quadratic equation with no \(b\) term and \(c=0\). To solve for \(s\), we need to perform an inverse operation to 'undo' the squaring. This process involves taking the square root of both sides of the equation. However, it's important to note that a square root has two values, a positive and a negative. In the context of physical problems like wind speed, we dismiss the negative root because wind speed cannot be negative.

To make it easier to understand, remember the square root is essentially asking: 'What number, multiplied by itself, will give me the original number?' This question is what you're answering when you're solving for \(s\) in quadratic equations such as the wave height equation.

Mathematical Modeling

Mathematical modeling is an incredibly powerful tool in translating real-world situations into mathematical terms. It involves creating equations to represent relationships between different quantities. In the example of the wave height modeling exercise, an equation \(h = 0.019s^2\) has been developed to express how the height of ocean waves (\(h\)) is dependent on the wind speed (\(s\)).

Through modeling, we can use algebra to predict and compare outcomes effectively. For instance, when tasked with finding the wind speeds that generate 5-foot and 20-foot waves, the model conveniently allows us to input these heights and calculate the respective necessary speeds. This not only demonstrates the utility of algebra in practical scenarios but also emphasizes the importance of a strong foundation in understanding how to manipulate algebraic expressions. Modeling gives us the language to quantify and solve problems that, on the surface, may not seem mathematical at all.

Square Roots in Algebra

Understanding square roots is essential when working with quadratic equations, especially when direct squaring or square terms are involved. The square root function is the inverse of the square function, meaning if \(x^2 = y\), then \(\sqrt{y} = x\), provided that \(x\) is non-negative.

In algebra, taking the square root is a common step when solving equations for a variable that's been squared. For example, in our ocean wave height problem, once we have isolated the square term \(s^2\), the next step is to find the square root to determine \(s\). However, it's important to remember that the square root of a number includes both a positive and negative value. Since practical situations often call for a physical quantity that cannot be negative, like time, distance, or in our case, wind speed, we select the positive square root as the meaningful solution.

When teaching or learning algebra, visual aids such as number lines or the construction of squares can offer a tangible understanding of the square root concept. Along with this, emphasizing the understanding that every positive number actually has two square roots can build a stronger mathematical foundation.