Let Jane's height last year be \( j \) cm. Then, Peter's height last year would be \( j + 5 \) cm since Peter was taller than Jane by 5 cm.
This year, Jane grew by 5 cm, so her height this year is: \[ j + 5 \ \text{cm} \]
Peter grew by 8 cm, so his height this year is: \[ (j + 5) + 8 = j + 13 \ \text{cm} \]
The total height of Peter and Jane this year is given as 282 cm. Therefore, we set up the equation: \[ (j + 5) + (j + 13) = 282 \]
Simplifying the left side: \[ 2j + 18 = 282 \]
Next, we isolate \( j \) by first subtracting 18 from both sides: \[ 2j = 282 - 18 \] \[ 2j = 264 \]
Now, divide by 2: \[ j = 132 \]
Thus, Jane's height last year was 132 cm. To find Jane's height this year, we add the growth she experienced: \[ 132 + 5 = 137 \ \text{cm} \]
Therefore, Jane's height this year is: \[ \boxed{137} \ \text{cm} \]