Answer:
 
And if we solve for a we got
 
So the value of height that separates the bottom 85% of data from the top 15% is 25.27.  
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  
Solution to the problem
Let X the random variable that represent the finishing time of a population, and for this case we know the distribution for X is given by:
 
  
Where  and
 and  
Solution to the problem
For this part we want to find a value a, such that we satisfy this condition:
 (a)
   (a)
 (b)
   (b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  
As we can see on the figure attached the z value that satisfy the condition with 0.85 of the area on the left and 0.15 of the area on the right it's z=1.04. On this case P(Z<1.04)=0.85 and P(z>1.04)=0.15
If we use condition (b) from previous we have this:
 
  
 
But we know which value of z satisfy the previous equation so then we can do this:
 
And if we solve for a we got
 
So the value of height that separates the bottom 85% of data from the top 15% is 25.27.