If
𝐺
is a function so that
𝐺
(
𝑚
)
represents the amount of gas (in gallons) used for driving
𝑚
miles, which one of the following represents the average gallons per mile for a distance of 5 miles?

Question

If 
𝐺
 is a function so that 
𝐺
(
𝑚
)
 represents the amount of gas (in gallons) used for driving 
𝑚
 miles, which one of the following represents the average gallons per mile for a distance of 5 miles?

BlackTom AI · Accepted Answer

We start by restating the problem to ensure clarity: G(m) gives gallons used for driving m miles, and we want the average gallons per mile for a distance of 5 miles.

Option 1: G(m+5) − G(m) / 5
This expression is ambiguous due to the lack of parentheses. It could be interpreted as G(m+5) minus [G(m) divided by 5], which would not represent an average rate over the interval [m, m+5]. The intended meaning in math notation is unclear here, so this option risks misinterpretation and does not reliably compute the average gallons per mile.

Option 2: G(m+5) − G(5) / m
This option compares G at m+5 with G at 5, and divides by m. It mixes two different starting points (m vs 5) and uses division by m, which does not yield the average rate over a 5-mile interval. This is inconsistent with the concept of average rate over a fixed distance starting from m, so it is incorrect.

Option 3: (G(m+5) − G(m)) / 5
This is the standard formula for average rate of change of G over an interval of length 5 miles: you take the change in gallons (G(m+5) − G(m)) and divide by the distance traveled (5 miles). This directly matches the definition of average gallons per mile over the 5-mile segment starting at m.

Option 4: G(m+5) / m + 5
This expression splits into G(m+5) divided by m and then adds 5, which has no basis in the concept of average gallons per mile. It does not compute a rate of change over the 5-mile interval and mixes units improperly, so it is incorrect.

Option 5: G(m+5) / 5
This option takes the gallons at distance m+5 and divides by 5, but it ignores the gallons used in the initial part (at m) and thus does not measure the average rate of gallons per mile over the 5-mile interval. It only considers a single endpoint, not the change over the interval.

In summary, the correctly defined average gallons per mile over a 5-mile segment from m to m+5 is obtained by the change in G over that interval, divided by the interval length, i.e., (G(m+5) − G(m)) / 5. The other choices fail to capture this standard average-rate-with-respect-to-distance concept in one of these ways: misinterpretation due to missing parentheses, mixing endpoints, or performing an improper division that does not reflect the average rate over the 5-mile interval.

If 𝐺 is a function so that 𝐺 ( 𝑚 ) represents the amount of gas (in gallons) used for driving 𝑚 miles, which one of the following represents the average gallons per mile for a distance of 5 miles?单项选择题

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