Let Jane's height last year be \( j \) cm. According to the information provided:
- Peter's height last year was \( j + 5 \) cm (since he was 5 cm taller than Jane).
This year, Peter grew by 8 cm and Jane grew by 5 cm. Therefore, their heights this year are:
- Peter's height this year: \( (j + 5) + 8 = j + 13 \) cm
- Jane's height this year: \( j + 5 \) cm
The total height of both Peter and Jane this year is given as 282 cm. Thus, we can set up the equation:
\[ (j + 13) + (j + 5) = 282 \]
Simplifying the equation:
\[ 2j + 18 = 282 \]
Now, subtract 18 from both sides:
\[ 2j = 282 - 18 \] \[ 2j = 264 \]
Now divide by 2:
\[ j = 132 \]
So, Jane's height last year was 132 cm. To find Jane's height this year, we now add the growth of 5 cm:
\[ j + 5 = 132 + 5 = 137 \]
Therefore, Jane’s height this year is \( \boxed{137} \) cm.