Problem 5 - Algebra 2

Question

The intensity of light emitted by a firefly can be determined by \(L(t)=10+0.3 t+0.4 t^{2}-0.01 t^{3},\) where \(t\) is temperature in degrees Celsius and \(L(t)\) is light intensity in lumens. If the temperature is \(30^{\circ} \mathrm{C},\) find the light intensity.

Step-by-Step Solution

Answer

The light intensity at 30°C is 109 lumens.

Step 1: Understand the Function

The function given is \(L(t) = 10 + 0.3t + 0.4t^2 - 0.01t^3\). This equation represents the relationship between the temperature \(t\) in degrees Celsius and the light intensity \(L(t)\) in lumens emitted by a firefly.

Step 2: Substitute the Temperature Value

You are asked to find the light intensity when the temperature \(t\) is 30°C. Substitute \(t = 30\) into the function: \[L(30) = 10 + 0.3(30) + 0.4(30)^2 - 0.01(30)^3\]

Step 3: Calculate Each Term

Next, calculate each term individually: 1. \(0.3(30) = 9\)2. \(0.4(30)^2 = 0.4 \times 900 = 360\)3. \(-0.01(30)^3 = -0.01 \times 27000 = -270\)

Step 4: Sum the Terms

After calculating each term, sum them to find \(L(30)\): \[L(30) = 10 + 9 + 360 - 270\]

Step 5: Find the Result

Add the numbers together: \(10 + 9 = 19\)\(19 + 360 = 379\)\(379 - 270 = 109\)Thus, the light intensity is 109 lumens.

Key Concepts

Temperature-Dependence

The concept of temperature-dependence highlights how certain properties or behaviors change with temperature. In our case, the light intensity emitted by a firefly is dependent on the temperature in degrees Celsius. As the temperature varies, the intensity of light changes according to the given polynomial function. Understanding temperature-dependence is crucial as:

• It demonstrates the relationship between two variables - here, temperature and light intensity.
• This knowledge can help predict how the firefly's light emission might change if the temperature changes.
• It provides insight into broader applications, such as why certain materials emit more light at different temperatures.

Thus, using the function provided helps quantify this relationship precisely for a specific temperature input.

Light Intensity

Light intensity refers to the brightness or the lumen output from a light source. In this scenario, the intensity emitted by a firefly is calculated in lumens based on the temperature. The equation given in the exercise relates temperature to light intensity, suggesting that as the temperature changes, so does the intensity. This is a real-world application of physics and biology working together. Other important points about light intensity include:

• Light intensity is critical for visibility and effectiveness in various settings, including environmental and biological studies.
• Understanding how temperature affects light intensity can inform studies on ecological and environmental conditions where fireflies thrive.
• It helps in mapping out behavioral patterns of temperature-sensitive creatures like the firefly.

Hence, by inputting the temperature into the provided polynomial function, we can determine the precise intensity level of light.

Algebraic Substitution

Algebraic substitution is a fundamental technique where you replace a variable with a given number to solve an equation. In this exercise, we substitute the temperature value into the polynomial to find the light intensity at that specific temperature. This method allows us to:

• Solve polynomial functions easily by simplifying them step by step.
• Evaluate real-world phenomena using mathematical models, like relating temperature and light intensity here.
• Provide immediate solutions to variable-dependent equations, which is useful in scientific and engineering contexts.

By substituting the temperature value 30°C into the polynomial, we can easily calculate the firefly's light intensity under this condition.

Degree of a Polynomial

The degree of a polynomial indicates the highest power of the variable in the polynomial function. In the provided function, the degree is 3, since the highest power of the variable (temperature \( t \)) is 3, in the term \(-0.01t^3\).Key aspects of understanding polynomial degrees include:

• The degree helps determine the general shape and behavior of the polynomial graph. In this case, a cubic polynomial can have more than one turning point, which represents changes in light intensity over varying temperatures.
• Higher-degree polynomials can model more complex relationships between variables, offering deeper insights into phenomena like temperature-dependence.
• The degree can also give us an idea of how many solutions or roots an equation may have.

Realizing the degree's significance allows us to comprehend possible outcomes and changes resulting from different temperature levels in the equation provided.