How do I use these to design a simple RLC circuit that does what the
manual tuner does at the middle of 10M?

Hi Ken, If you have the equipment to determine the impedance of the antenna at
the middle of the 10M band, there are a number of ways to match it to 50 ohms.

First of all, you don't want an RLC circuit. Leave out the R cause that is
power lost as heat. You want everything to be radiated. An "L" network
consisting of an inductor and capacitor will match anything to anything.
What I usually do is determine the feedpoint impedance of the antenna at the
frequency of interest, and use the Smith Chart to determine the configuration
and values of an "L" network to get a match.
If you are not comfortable with the Chart, there are some programs on the net
that will design the network for you. Or, if you give me the complex impedance
and frequency, I will be glad to provide a network.
73 Gary N4AST