Let's break down the given problem step by step.
### Part i: Deriving the Formula
We need to find a formula that relates yesterday's closing price (denoted as [tex]\( Y \)[/tex]) and today's closing price (denoted as [tex]\( T \)[/tex]). According to the problem, the stock rose by [tex]\( 1 \frac{5}{8} \)[/tex] points today.
Starting from yesterday's closing price [tex]\( Y \)[/tex], we add the rise in points to get today's closing price [tex]\( T \)[/tex]. Therefore, the formula should be:
[tex]\[ T = Y + 1 \frac{5}{8} \][/tex]
### Part ii: Calculating Today's Closing Price
We are given that yesterday's closing price was [tex]\( 32 \frac{13}{16} \)[/tex].
First, let's convert the fractions to decimals to make the arithmetic easier:
- Yesterday's closing price:
[tex]\[
32 \frac{13}{16} = 32 + \frac{13}{16} = 32 + 0.8125 = 32.8125
\][/tex]
- Points gained today:
[tex]\[
1 \frac{5}{8} = 1 + \frac{5}{8} = 1 + 0.625 = 1.625
\][/tex]
Now, let's add the rise in points to yesterday's closing price to find today's closing price:
[tex]\[
T = 32.8125 + 1.625 = 34.4375
\][/tex]
So today's closing price is [tex]\( 34.4375 \)[/tex].
### Summary of the Answers
i) The formula that relates [tex]\( Y \)[/tex] and [tex]\( T \)[/tex] is:
[tex]\[ T = Y + 1 \frac{5}{8} \][/tex]
ii) If yesterday's closing price was [tex]\( 32 \frac{13}{16} \)[/tex], today's closing price is [tex]\( 34 \frac{7}{16} \)[/tex] (in fractional form, considering [tex]\( 34.4375 = 34 \frac{7}{16} \)[/tex]).
Thus, the correct options are:
i) [tex]\( T = Y + 1 \frac{5}{8} \)[/tex]
ii) Today's closing price was [tex]\( 34 \frac{7}{16} \)[/tex].
