In this question we have 4 items. Let's answer each one of them individually.
item (a):
To answer this question, first we need to convert our desired value into a z-score. For all of the items, we have the following mean(mu) and standard deviation(sigma):
[tex]\begin{cases}\mu=42.3 \\ \sigma=13.8\end{cases}[/tex]
The formula to convert a value from our distribution to a z-score is:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
With x = 57.5, we have the following z-score:
[tex]z=\frac{57.5-42.3}{13.8}\approx1.10[/tex]
Using a z-table(right), we can find the area to the right hand side of the curve. Those values represent the area between z = 0 and any positive value, we just need to find the area between z = 0 to our z-score.
According to this z-table, the area between 42.3 gallons and 57.5 gallons is 36.43% of the whole area. Since the normal distribution is symmetric, from the mean 42.3 gallons to the left we have 50% of the graph. Then, the probability of having the consumption below 57.5 is the sum of those two values.
[tex]0.5+0.3643=0.8643[/tex]
Then, this is our answer.
[tex]P(x<57.5)=0.8643[/tex]
item (b):
