Question
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.)
Between
(Round to three decimal places as needed.)
Between
(Round to three decimal places as needed.)
Ask by Mcdonald Rogers. in the United States
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Between
and
, the train travels 9.091 miles.
Solution

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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.