What is the range of D(m)?

Understand the Problem

The question is asking for the range of the function D(m), which represents the distance Mariana has left to run after m minutes in a 10-mile race. We need to determine the valid output values of D(m) based on her running distance over time.

Answer

The range of $ D(m) $ is all real numbers from 0 to 10.

Steps to Solve

Understand the total distance and time Mariana is running a 10-mile race, which means initially, the distance left to run is 10 miles. The race is expected to finish in 90 minutes.
Determine the distance left as a function of time The distance ( D(m) ) in miles after ( m ) minutes can be represented as: $$ D(m) = 10 - \frac{m}{9} $$ Here, ( \frac{m}{9} ) represents the distance covered in miles at a pace of 9 minutes per mile.
Determine the range of ( D(m) )

At ( m = 0 ) (start), ( D(0) = 10 ) miles.
At ( m = 90 ) (end), ( D(90) = 0 ) miles.

Thus, as ( m ) increases from 0 to 90, ( D(m) ) will decrease from 10 to 0.

Contextualize the range Since ( D(m) ) decreases from 10 to 0, the valid output values for ( D(m) ) are all numbers from 0 to 10, inclusive.

The range of ( D(m) ) is all real numbers from 0 to 10.

More Information

Since Mariana's distance covered increases continuously as time passes, every fractional distance from 0 to 10 is possible, making the range include all real numbers in that interval.

Tips

Misunderstanding the range to be from 0 to 90 instead of 0 to 10.
Forgetting that the distance should decrease as time increases.

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