To solve for [tex]\( r \)[/tex] given the formula [tex]\( A = P(1 + r t) \)[/tex] with [tex]\( A = 27 \)[/tex] and [tex]\( t = 3 \)[/tex], let's break down the problem step by step.
First, we substitute the given values into the formula:
[tex]\[ 27 = P(1 + r \cdot 3) \][/tex]
Now, solve for [tex]\( r \)[/tex]:
1. Start with the original equation:
[tex]\[ 27 = P(1 + 3r) \][/tex]
2. Isolate the term containing [tex]\( r \)[/tex]:
[tex]\[ \frac{27}{P} = 1 + 3r \][/tex]
3. Subtract 1 from both sides:
[tex]\[ \frac{27}{P} - 1 = 3r \][/tex]
4. To simplify, represent [tex]\( \frac{27}{P} - 1 \)[/tex] as a single fraction:
[tex]\[ \frac{27 - P}{P} = 3r \][/tex]
5. Solve for [tex]\( r \)[/tex] by dividing both sides by 3:
[tex]\[ r = \frac{27 - P}{3P} \][/tex]
After working through these steps, we reach the expression for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{27 - P}{3P} \][/tex]
Now, compare this with the given options:
A. [tex]\( r = \frac{27 - 1}{3P} \)[/tex]
B. [tex]\( r = \frac{27 - P}{3} \)[/tex]
C. [tex]\( r = \frac{27 - P}{3P} \)[/tex]
D. [tex]\( r = \frac{26}{3} \)[/tex]
The correct answer is clearly Option C:
[tex]\[ \boxed{r = \frac{27 - P}{3P}} \][/tex]
