Question

theres a roughly linear relationship between the number of times a species of cricket will chirp in one minute and the temperature outside. for a certain type of cricket, this relationship can be expressed using the formula ( t = 0.27c + 40 ), where ( t ) represents the temperature in degrees fahrenheit and ( c ) represents the number of times the cricket chirps in one minute. what is the meaning of the ( c )-value when ( t = 76 )?

answer

• the number of times the cricket could be expected to chirp in one minute if it’s 76° f.
• the change in temperature in degrees fahrenheit for each additional cricket chirp in one minute.
• the expected temperature in degrees fahrenheit if the cricket has chirped 0 times per minute.
• the expected temperature in degrees fahrenheit if the cricket has chirped 76 times per minute.

Explanation:

Step1: Start with the given formula

We have the formula \( T = 0.27c + 40 \), and we know that \( T = 76 \). So we substitute \( T \) with 76 in the formula:
\( 76 = 0.27c + 40 \)

Step2: Subtract 40 from both sides

To isolate the term with \( c \), we subtract 40 from both sides of the equation:
\( 76 - 40 = 0.27c + 40 - 40 \)
\( 36 = 0.27c \)

Step3: Solve for \( c \)

Now we divide both sides by 0.27 to find the value of \( c \):
\( c=\frac{36}{0.27} \)
\( c = 133.33\cdots\) (but we are looking for the meaning, not the exact calculation here. Wait, the question is about the meaning of \( c \) when \( T = 76 \). Let's re - interpret. The formula is \( T = 0.27c+40 \), where \( T \) is temperature and \( c \) is chirps per minute. When \( T = 76 \), we are solving for \( c \), which represents the number of chirps per minute when the temperature is 76 degrees Fahrenheit. Wait, let's check the options. The options are about the meaning of \( c \) when \( T = 76 \). So we solve \( 76=0.27c + 40 \) for \( c \). Let's do the algebra correctly.

First, \( T = 0.27c+40 \), solve for \( c \) when \( T = 76 \):

Subtract 40: \( 76 - 40=0.27c\) => \( 36 = 0.27c\)

Then \( c=\frac{36}{0.27}=\frac{36\times100}{27}=\frac{3600}{27}=\frac{400}{3}\approx133.33 \)

But the question is about the meaning of \( c \) when \( T = 76 \). So \( c \) is the number of chirps per minute when the temperature \( T \) is 76 degrees Fahrenheit. Looking at the options, the correct interpretation is "The number of times the cricket could be expected to chirp in one minute if it's 76°F" (wait, no, let's check the options again. Wait the options:

1. The number of times the cricket could be expected to chirp in one minute if it's 76°F.

1. The change in temperature in degrees Fahrenheit for each additional cricket chirp in one minute.

1. The expected temperature in degrees Fahrenheit if the cricket has chirped 0 times per minute.

1. The expected temperature in degrees Fahrenheit if the cricket has chirped 76 times per minute.

Wait, we have the formula \( T = 0.27c + 40 \). So \( T \) is a function of \( c \). When we set \( T = 76 \) and solve for \( c \), we are finding the value of \( c \) (chirps per minute) that corresponds to a temperature of 76 degrees Fahrenheit. So the meaning of \( c \) when \( T = 76 \) is the number of chirps per minute when the temperature is 76°F, which is the first option. But let's confirm with the algebra.

We start with \( T = 0.27c + 40 \). We want to find \( c \) when \( T = 76 \). So:

\( 76=0.27c + 40 \)

Subtract 40: \( 36 = 0.27c \)

Divide by 0.27: \( c=\frac{36}{0.27}\approx133.33 \)

So \( c \) here represents the number of chirps per minute when the temperature is 76°F. So the correct option is "The number of times the cricket could be expected to chirp in one minute if it's 76°F".

Answer:

The number of times the cricket could be expected to chirp in one minute if it's 76°F.