Isoquant and Isocost

A student posted the following question:

"The owner of a small car-rental service is trying to decide on the appropriate numbers of vehicles and mechanics to use in the business for the current level of operations. He recognizes that his choice represents a trade-off between the two resources. His past experience indicates that this trade-off is as follows:
Vehicles:     100  70  50  40  35  32
Mechanics:  2.5  5   10  15   25   35
a] Assume that annual (leasing) cost per vehicle is $6,000 and the annual salary per mechanic is $25,000. What combination of vehicles and mechanics should he employ? - Already got the answer to which is 70 vehicles and 5 mechanics
b] Illustrate this problem with the use of an isoquant/isocost diagram. Indicate graphically the optimal combination of resources.

Thank you for your help and time"

My response:

Your goal here should be not just to answer the question but more importantly to gain some understanding for why the answer is correct.  To do that you should solve this both numerically and graphically.

Numerically - The various combinations of points given lie on the same isoquant.  There is economics in computing the slope of the segment between consecutive points.  The absolute value of that slope is called the rate of technical substitution.  (Sometimes the word marginal precedes that so the entire expression is marginal rate of technical substitution.)  You should compute the full schedule of RTS.

Then you should compute the ratio of the input prices.  Students doing this for the first time are unsure whether that ratio should be (price vehicles)/(price mechanics) or (price mechanics)/(price vehicles).  The graphical approach should help there.  Which input is on the x axis?  the y axis?  Knowing that you can plot a line of constant expenditure on input bundles.  The input price ratio you want is the absolute value of the slope of the line.

The economics is in understanding when the RTS does not equal the relative price:  movement in which direction along the isoquant will result in lower cost?

Graphically -  You should plot the isoquant.  In the same diagram you should plot several isocost lines including one that lies entirely below the isoquant and another that crosses the isoquant through a non-optimal input bundle.   If at that crossing point the isoquant is steeper, which direction along the isoquant leads to lower cost.  Your goal here is to tie the graphical approach to the numerical approach.  They are different representations of the same idea.

Only after doing the above should you plot the isocost through the cost minimizing bundle.  The conditions that characterize the optimum are best understood as being when the conditions for a non-optimal bundle don't hold.