Problem 71 - Algebra and Trigonometry

Question

Crickets and Temperature Biologists have observed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 120 chirps per minute at \(70^{\circ} \mathrm{F}\) and 168 chirps per minute at \(80^{\circ} \mathrm{F} .\) (a) Find the linear equation that relates the temperature \(t\) and the number of chirps per minute \(n\) . (b) If the crickets are chirping at 150 chirps per minute, estimate the temperature.

Step-by-Step Solution

Answer

(a) The equation is \(n = 4.8t - 216\). (b) Estimated temperature is \(76.25^{\circ}\)F.

Step 1: Determine the Slope of the Line

The relationship between chirps and temperature is linear, and the slope \(m\) can be calculated using the two given points: \((70, 120)\) and \((80, 168)\). The formula for the slope \(m\) is \(m = \frac{n_2 - n_1}{t_2 - t_1}\). Substitute the values to find \(m = \frac{168 - 120}{80 - 70} = \frac{48}{10} = 4.8\).

Step 2: Determine the Linear Equation

Now that we have the slope, use the point-slope form of a linear equation which is \(n - n_1 = m(t - t_1)\). We'll use the point \((70, 120)\): \(n - 120 = 4.8(t - 70)\). Simplify to find the equation: \(n = 4.8t - 336 + 120\), which further simplifies to \(n = 4.8t - 216\).

Step 3: Substitute to Estimate the Temperature

With the equation \(n = 4.8t - 216\), substitute \(n = 150\) to estimate the temperature: \(150 = 4.8t - 216\). Rearrange to solve for \(t\):\[ 150 + 216 = 4.8t \]\[ 366 = 4.8t \]\[ t = \frac{366}{4.8} \approx 76.25 \]

Step 4: Verify the Linear Model

Plotting or checking the model can further ensure accuracy. Calculate chirps at known temperatures, 70°F and 80°F, using the equation \(n = 4.8t - 216\): - For 70°F, \(n = 4.8 \times 70 - 216 = 120\).- For 80°F, \(n = 4.8 \times 80 - 216 = 168\).These match the problem, verifying our equation is correct.

Key Concepts

Slope Calculation

When analyzing the relationship between two variables like chirps and temperature, the first step is often finding the slope of the line. This step is crucial because it indicates how one variable changes concerning another.

The slope (\(m\)) is computed using the formula:

• \(m = \frac{n_2 - n_1}{t_2 - t_1}\)

Here, \((n_1, t_1)\) and \((n_2, t_2)\) are the points representing the pair of chirps and temperature. These points are derived from actual observations, which in this case are \((70, 120)\) and \((80, 168)\).
Substituting these values, the calculation becomes:

• \(m = \frac{168 - 120}{80 - 70} = \frac{48}{10} = 4.8\)

This slope of \(4.8\) tells us that for each degree Fahrenheit increase in temperature, cricket chirps increase by 4.8 chirps per minute. Such insights are fundamental for making predictions about how the chirping rate will change as temperature varies.

Point-Slope Form

After determining the slope, the next step in forming a linear equation is to use the point-slope form. This form is handy when you know a point on the line and its slope.

The point-slope form formula is:

• \(n - n_1 = m(t - t_1)\)

In applying this, we take one of our known points \((70, 120)\) and the slope \(4.8\) to form the equation:

• \(n - 120 = 4.8(t - 70)\)

By expanding and simplifying, we obtain the linear equation:

• \(n = 4.8t - 336 + 120\)
• \(n = 4.8t - 216\)

This equation captures the relationship between the chirping rate \(n\) and temperature \(t\). It allows us to calculate chirps per minute for any given temperature. Understanding this form enables modeling real-world phenomena with simplicity and precision.

Temperature Estimation

Using our derived linear equation, it’s possible to estimate the temperature based on observed chirping rates. Suppose crickets are chirping at 150 chirps per minute, and you aim to find the ambient temperature.

Substitute \(n = 150\) into the equation \(n = 4.8t - 216\):

• \(150 = 4.8t - 216\)

Rearrange the equation to solve for \(t\):

• \(150 + 216 = 4.8t\)
• \(366 = 4.8t\)
• \(t = \frac{366}{4.8} \approx 76.25\)

The estimated temperature is approximately \(76.25^{\circ}F\). Applying this equation helps in situations where direct temperature measurement is challenging, allowing you to infer temperature from cricket behavior.

Linear Model Verification

Lastly, verifying a linear model ensures its reliability and consistency with the data. This process often involves checking whether the equation predicts known data points accurately.

For this model, use the equation \(n = 4.8t - 216\) to verify the chirping rates at our initial temperatures 70°F and 80°F:
At 70°F:

• \(n = 4.8 \times 70 - 216 = 120\)

At 80°F:

• \(n = 4.8 \times 80 - 216 = 168\)

These calculations perfectly match our provided data points of 120 chirps per minute at 70°F and 168 chirps per minute at 80°F. This confirms that our linear model accurately reflects the relationship between temperature and chirping rate for this species of cricket. Verifying models is crucial in practical applications, as it lends confidence to predictions made using the model.