a) 103.176 m / min
b) 1751.28 meters
Explanation:
Given:-
- Emma's and Lily's velocities ( E(t) and L(t) ) are given as functions respectively:
- Where, E ( t ) and L ( t ) are given in m / min
- Both run for a total time of 15 minutes in the same direction along the straight track defined by the absolute interval:
( 0 ≤ t ≤ 15 ) mins
- It is known that Emma is 10 meters ahead of Lily at time t = 0.
Find:-
a) Find the value of using correct units, interpret the meaning of
b) What is the maximum distance between Emma and Lily over the time interval 0 ≤ t ≤ 15?
Solution:-
- The average value of a function f ( x ) over an interval [ a , b ] is determined using calculus via the following relation:
- The first part of the question is asking us to determine the average velocity of Emma over the time interval of ( 2 , 8 ). Therefore, ( E_avg ) can be determined using the above relation:
- We will evaluate the integral formulation above to determine Emma's average velocity over the 2 ≤ t ≤ 8 minute range:
- Complete the square in the denominator:
- Use the following substitution:
- Substitute the relations for (u) and (dt) in the above E_avg expression.
- Use the following standard integral:
- Evaluate:
- Apply back substitution for ( u ):
- Plug in the limits and find Emma's average velocity:
Emma's average speed over the interval ( 2 ≤ t ≤ 8 ) is 103.179 meters per minute.
- The displacement S ( E ) of Emma from time t = 0 till time ( t ) over the absolute interval of 0≤t≤15 is given by the relation:
- The displacement S ( L ) of Lily from time t = 0 till time ( t ) over the absolute interval of 0 ≤ t ≤ 15 is given by the relation:
Apply integration by parts:
Re-apply integration by parts 2 more times:
- The distance between Emma and Lily over the time interval 0 < t < 15 mins can be determined by subtracting S ( L ) from S ( E ):
- The maximum distance ( S ) between Emma and Lily is governed by the critical value of S ( t ) for which function takes either a minima or maxima.
- To determine the critical values of the function S ( t ) we will take the first derivative of the function S with respect to t and set it to zero:
- We will plug in each value of t and evaluate the displacement function S(t) for each critical value: