Answer:
a)  
 
b)  
 
c)  
 
Sicne this probability just represent 1% of the data we can consider this value as unusual.
d)  
And if we solve for a we got
 
So the value of height that separates the bottom 90% of data from the top 10% is 87360.  
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".  
Part a
Let X the random variable that represent the life of a particular auto transmission of a population, and for this case we know the distribution for X is given by:
 
  
Where  and
 and  
We are interested on this probability
 
And the best way to solve this problem is using the normal standard distribution and the z score given by:
 
If we apply this formula to our probability we got this:
 
And we can find this probability using excel or the normal standard table and we got:
 
Part b
 
And the best way to solve this problem is using the normal standard distribution and the z score given by:
 
If we apply this formula to our probability we got this:
 
And we can find this probability using the complement rule and excel or the normal standard table and we got:
 
Part c
 
And the best way to solve this problem is using the normal standard distribution and the z score given by:
 
If we apply this formula to our probability we got this:
 
And we can find this probability using the complement rule and excel or the normal standard table and we got:
 
Sicne this probability just represent 1% of the data we can consider this value as unusual.
Part d
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 0.9 of the area on the left and 0.1 of the area on the right it's z=1.28. On this case P(Z<1.28)=0.9 and P(z>1.28)=0.1
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 90% of data from the top 10% is 87360.