Question
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At , a train approaching a station begins decelerating from a speed of according to the acceleration function
, where . How far does the train travel between and ? Between and ?
Between and the train travels 10.000 mi .
(Round to three decimal places as needed.)
Between and the train travels .
(Round to three decimal places as needed.)

Ask by Mcdonald Rogers. in the United States
Jan 23,2025

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Answer

Between and , the train travels 9.091 miles.

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Mind Expander

To find the distance the train travels between and , we first need to find the velocity function. We can integrate the acceleration function to get the velocity.
Given that the acceleration , we integrate with respect to :
Using substitution gives or . The limits of integration will change accordingly:
If , .
If , .
Thus, the integral becomes:
Now, we apply the initial condition. At , we have :
So, the velocity function is:
Next, we find the distance traveled between and :
Using the substitution :
When , and when , . The differential becomes :
Calculating the above term:
Thus, between and , the train travels approximately mi.

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