Question
Today, Howell has
\[75\] kiwis. He will eat
\[2\] of them every day.
Is the number of kiwis Howell has left proportional to the number of days that pass?
\[75\] kiwis. He will eat
\[2\] of them every day.
Is the number of kiwis Howell has left proportional to the number of days that pass?
Answers
Answered by
GPT-4o mini
To determine if the number of kiwis Howell has left is proportional to the number of days that pass, let's define a few variables:
- Let \( K(d) \) be the number of kiwis Howell has left after \( d \) days.
- He starts with \( 75 \) kiwis and eats \( 2 \) kiwis each day.
After \( d \) days, the number of kiwis Howell has left is given by the equation:
\[
K(d) = 75 - 2d
\]
To assess whether the number of kiwis is proportional to the number of days, we need to see if \( K(d) \) can be expressed as a constant multiple of \( d \).
For two quantities to be proportional, there must exist a constant \( k \) such that:
\[
K(d) = k \cdot d
\]
However, from our equation \( K(d) = 75 - 2d \), we can see that it is not expressed in the form of \( k \cdot d \) because it includes a constant term (75). Instead, it is a linear equation with a negative slope (due to the \(-2d\) term) and an intercept (75).
As \( d \) increases:
- The value of \( K(d) \) decreases linearly until it reaches \( 0 \) (or below).
- The relationship between the number of kiwis left and the number of days is not of the form \( K(d) = k \cdot d \) or a direct proportionality \( K(d) = k \cdot d + c\), where \( c \) is any constant.
Thus, **the number of kiwis Howell has left is not proportional to the number of days that pass**. Instead, it is a linear function that decreases with increasing days but does not maintain a constant ratio to the days.
- Let \( K(d) \) be the number of kiwis Howell has left after \( d \) days.
- He starts with \( 75 \) kiwis and eats \( 2 \) kiwis each day.
After \( d \) days, the number of kiwis Howell has left is given by the equation:
\[
K(d) = 75 - 2d
\]
To assess whether the number of kiwis is proportional to the number of days, we need to see if \( K(d) \) can be expressed as a constant multiple of \( d \).
For two quantities to be proportional, there must exist a constant \( k \) such that:
\[
K(d) = k \cdot d
\]
However, from our equation \( K(d) = 75 - 2d \), we can see that it is not expressed in the form of \( k \cdot d \) because it includes a constant term (75). Instead, it is a linear equation with a negative slope (due to the \(-2d\) term) and an intercept (75).
As \( d \) increases:
- The value of \( K(d) \) decreases linearly until it reaches \( 0 \) (or below).
- The relationship between the number of kiwis left and the number of days is not of the form \( K(d) = k \cdot d \) or a direct proportionality \( K(d) = k \cdot d + c\), where \( c \) is any constant.
Thus, **the number of kiwis Howell has left is not proportional to the number of days that pass**. Instead, it is a linear function that decreases with increasing days but does not maintain a constant ratio to the days.