Question

The chirping rate of a cricket, \(\mathrm{X}\), in chirps per minute, near room temperature is $$ \mathrm{X}=7.2 t-32 $$ where \(t\) is the temperature in \({ }^{\circ} \mathrm{C}\). (a) Calculate the chirping rates at \(25^{\circ} \mathrm{C}\) and \(35^{\circ} \mathrm{C}\). (b) Use your answers in (a) to estimate the activation energy for the chirping. (c) What is the percentage increase for a \(10^{\circ} \mathrm{C}\) rise in temperature?

Short answer

Answer: The percentage increase in the chirping rate of the cricket for a 10°C rise in temperature is approximately 48.65%.

Step-by-step solution

Calculate the chirping rates at 25°C and 35°C

Plug in the values of 25°C and 35°C into the given formula X = 7.2t - 32 to find the chirping rates. For 25°C: X = 7.2 * 25 - 32 For 35°C: X = 7.2 * 35 - 32

Calculate the chirping rates

Solve the equations obtained in step 1 to find the chirping rates at 25°C and 35°C. For 25°C: X = 7.2 * 25 - 32 = 180 - 32 = 148 (chirps per minute) For 35°C: X = 7.2 * 35 - 32 = 252 - 32 = 220 (chirps per minute)

Estimate the activation energy for chirping

We cannot directly calculate the activation energy for chirping using the provided information in the problem. However, we can observe that the activation energy is related to the temperature difference required to produce a noticeable change in the chirping rate. A larger activation energy would require a larger temperature change to yield a noticeable difference in the chirping rate. Comparing the chirping rates at 25°C and 35°C will give us an idea of how temperature affects the chirping rate.

Calculate the percentage increase in chirping rate for a 10°C rise in temperature

To find the percentage increase in chirping rate for a 10°C rise in temperature, we will use the formula: Percentage Increase = ((\(X_{2} - X_{1}) / X_{1}) * 100\)% where \(X_{1}\) is the initial chirping rate (at 25°C) and \(X_{2}\) is the final chirping rate (at 35°C). Percentage Increase = ((\(220 - 148) / 148) * 100\) %

Calculate the percentage increase

Solve the equation obtained in step 4 to find the percentage increase in chirping rate for a 10°C rise in temperature: Percentage Increase = ((\(220 - 148) / 148) * 100\) % = (72 / 148) * 100\( % = 48.65\) % So, the chirping rate of the cricket increases by approximately 48.65% for a 10°C rise in temperature.