Question

Use the following information. The power generated by a windmill can be modeled by the equation \(w=0.015 s^{3},\) where \(w\) is the power measured in watts and \(s\) is the wind speed in miles per hour. Find the ratio of the power generated by a windmill when the wind speed is 20 miles per hour to the power generated when the wind speed is 10 miles per hour.

Short answer

The ratio of power generated by the windmill when the wind speed is 20 mph to the power generated when it is 10 mph is 8:1.

Step-by-step solution

Compute Power at 20 mph

Substitute 20 into the power generation formula for \(s\). This would give \(w20 = 0.015 * (20)^3\).

Compute Power at 10 mph

Next, substitute 10 into the same formula. This would give \(w10 = 0.015 * (10)^3\).

Compute Ratio

To find the ratio asked for in the question, divide \(w20\) by \(w10\). This will give the final ratio.

Key concepts

Algebraic Modeling

Algebraic modeling is a powerful mathematical method for representing real-world situations with algebraic equations. In our windmill power generation scenario, such a model helps to relate wind speed to the power output of a windmill. The provided model, (w=0.015 s^{3}), is used to predict the power generated ((w)) by the windmill at different wind speeds ((s)).

The model is based on the assumption that the power output increases with the cube of the wind speed, which is a relationship derived from physical principles. In algebraic modeling, the key steps often include identifying the variables, establishing their relationships based on context (like physical laws), and forming an equation that can be used to compute specific values. Students can learn to adjust these models as more information becomes available or as conditions change.

Understanding algebraic modeling helps students approach complex problems by breaking them down into simpler, more manageable parts. Algebraic expressions and equations are the tools that enable this breakdown, making them fundamental to mastering algebra and applying mathematics in diverse fields.

Exponential Functions

Exponential functions model situations where a quantity grows or decays at a rate proportional to its current value. They are characterized by an equation of the form (y = ab^x), where (y) is the final amount, (a) is the initial amount, (b) is the base or growth factor, and (x) is the exponent.

In our example, the cube of the wind speed (s^3), represents an exponential relationship between the wind speed and the power generated because the power is proportional to the third power of the wind speed. Recognizing that the wind-power relationship follows an exponential pattern allows us to predict how small changes in wind speed can dramatically affect power output.

This is why it's important to understand exponential functions in real-world applications such as renewable energy production, where power output can be extremely sensitive to environmental conditions. While the windmill power equation is not a traditional exponential function with a constant base multiplied by a variable exponent, it conveys a similar concept of rapid growth.

Ratio and Proportion

The concepts of ratio and proportion involve comparing quantities to determine their relative sizes or to establish a consistent relationship between variables. In the case of windmill power generation, we're interested in the ratio of power outputs at two different wind speeds.

To find this ratio, we can use the formula provided to calculate the power generated at each speed and then form a quotient. This exercise explicitly asks us to compare the power at 20 mph to that at 10 mph. By calculating each power value and dividing them, we get the ratio (w20/w10), which conveys how much more power is generated at the higher wind speed compared to the lower.

Understanding ratios is critical for assessing the relative effectiveness or efficiency of processes and systems in various fields. For instance, recognizing how a windmill's power generation scales with wind speed can guide the optimal placement and design of wind farms, leading to better use of resources and increased energy production.