Key ideas to get started on a rate problem:

• If it's a problem involving rates write down the definition of the rate.
• Then identify everything you can find related to that definition.

If you know

• the rate r1 at which period is changing with respect to length at one length L1

and

• the rate r2 at which period is changing with respect to length at another length L2

then how do you find the change in the period?

We've got length and rate information.  We want to find change in period.

The rate is the rate at which period changes with respect to length.

Sorta confusing.  It helps greatly to analyze the phrase 'the rate at which period changes with respect to length'.

• the average rate at which period changes with respect to length is

ave rate = change in period / change in length.

Check the status of the three quantities in the problem:

• change in period: this is what we're trying to find, the unknown in the problem
• change in length:  length changes from L1 to L2 so change in length is L2 - L1
• ave rate:  we aren't given average rate but we are given two rates r1 and r2.  If we average the two rates we get a good approximation of the average rate.  The average of the two rates is (r1 + r2) / 2.

Using

• change in length = L2 - L1
• average rate = (r1 + r2) / 2

we can rewrite

• ave rate = change in period / change in length

to get

• change in period = ave rate * change in length

and substitute to get

• change in period = (r1 + r2) / 2 * (L2 - L1).