The lengths of two rods are 15.2 ± 0.2 cm and 10.7± 0.2cm. Find difference in lengths of the two rods with the limits of error.

To find the difference in lengths of the two rods with the limits of error, we can analyze the given values step-by-step.

Let the length of the first rod be $L_1$ and the length of the second rod be $L_2$:
$L_1=(15.2\pm 0.2)\text{cm}$
$L_2=(10.7\pm 0.2)\text{cm}$

Here, the measured values are:
$l_1=15.2\text{cm}$
$l_2=10.7\text{cm}$

And the absolute errors in these measurements are:
$\Delta l_1=0.2\text{cm}$
$\Delta l_2=0.2\text{cm}$

Step 1: Calculate the difference in the measured lengths
Let $L$ be the difference in the lengths:
$L=l_1-l_2$
$L=15.2-10.7=4.5\text{cm}$

Step 2: Calculate the limit of error for the difference
When two quantities are subtracted, their absolute errors are added to get the maximum possible error in the result. Therefore, the absolute error $\Delta L$ in the difference is:
$\Delta L=\Delta l_1+\Delta l_2$
$\Delta L=0.2+0.2=0.4\text{cm}$

Step 3: Write the final result with error limits
The difference in lengths is:
$L\pm \Delta L=(4.5\pm 0.4)\text{cm}$

Thus, the correct option is 4.5 ± 0.4 cm.