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The chirp rate, [tex]C[/tex], in chirps per minute, of the snowy tree cricket is given by [tex]C = f(T) = 4T - 160[/tex], where [tex]T[/tex] is degrees Fahrenheit.

(a) Find an appropriate domain of [tex]f[/tex] in the context of the model, assuming a maximum temperature of [tex]134^{\circ}F[/tex], the highest recorded at a weather station.

Domain of [tex]f[/tex] is [tex]\square \leq T \leq 134[/tex]

(b) Find the range of [tex]f[/tex] on this domain.

Range of [tex]f[/tex] is [tex]\square \leq C \leq \square[/tex]

Answer :

Let's approach this problem step by step.

### Part (a): Finding the Domain

The chirp rate function given is:
[tex]\[ C = f(T) = 4T - 160 \][/tex]

To find the domain of [tex]\( f \)[/tex], we need to determine the range of temperatures [tex]\( T \)[/tex] for which this function is valid. The context of the model states that the highest recorded temperature at a weather station is [tex]\( 134^{\circ} F \)[/tex].

1. Let's first find the lower limit of [tex]\( T \)[/tex]. For the chirp rate [tex]\( C \)[/tex] to make sense, it must be non-negative because a negative chirp rate is not physically meaningful:
[tex]\[ 4T - 160 \geq 0 \][/tex]

Solving this inequality for [tex]\( T \)[/tex]:
[tex]\[ 4T \geq 160 \][/tex]
[tex]\[ T \geq 40 \][/tex]

2. The upper limit of [tex]\( T \)[/tex] is given as [tex]\( 134^{\circ} F \)[/tex].

Putting this information together, the domain of [tex]\( f \)[/tex] is:
[tex]\[ 40 \leq T \leq 134 \][/tex]

### Part (b): Finding the Range

Next, we need to find the range of [tex]\( f \)[/tex] on the domain we found. Specifically, we need to calculate the chirp rates at the minimum and maximum temperatures in this domain.

1. For the minimum temperature [tex]\( T = 40 \)[/tex]:
[tex]\[ C = 4(40) - 160 = 160 - 160 = 0 \][/tex]

2. For the maximum temperature [tex]\( T = 134 \)[/tex]:
[tex]\[ C = 4(134) - 160 = 536 - 160 = 376 \][/tex]

Thus, the range of [tex]\( f \)[/tex] is:
[tex]\[ 0 \leq C \leq 376 \][/tex]

### Summary

- Domain of [tex]\( f \)[/tex]:
[tex]\[ 40 \leq T \leq 134 \][/tex]

- Range of [tex]\( f \)[/tex]:
[tex]\[ 0 \leq C \leq 376 \][/tex]

So, your answers are:

(a) Domain of [tex]\( f \)[/tex] is [tex]\( 40 \leq T \leq 134 \)[/tex].

(b) Range of [tex]\( f \)[/tex] is [tex]\( 0 \leq C \leq 376 \)[/tex].

Thanks for taking the time to read The chirp rate tex C tex in chirps per minute of the snowy tree cricket is given by tex C f T 4T 160 tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

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