Determine the 3-D planes whose point of intersection would solve the following problem:

Question

A car rental company spent $50,000,000 in putting together a fleet of small, medium, and largecars by purchasing them from three motor retail outlets in the form of packages that consistedof a mixture of different sized cars. MR1, the first motor retail outlet, offered packages costing$500,000 each that consisted of 5 small cars, 5 medium cars, and 10 large cars; MR2 offered packages costing $400,000 each that consisted of 10 small, 5 medium, and 5 large cars; andMR3 offered packages costing $200,000 each that consisted of 5 small cars (no medium cars)and 5 large cars.The rental company purchased exactly a third of its packages from MR3 andended up with exactly twice as many small cars as medium and large combined. How manypackages did the company purchase from each retail outlet?Clearly state what each of your variables denotes ... and just set up the equations, do not solvethe system.

Answer 1

Using A, B, and C for the number of packages purchased from MR1, MR2, and MR3, respectively, we have the first two planes:

 

50,000,000 = 500,000A + 400,000B + 200,000C

C = (1/3)(A + B + C)

 

Doing some work, s = small cars, m = medium cars, l = large cars;

s = 5A + 10B + 5C

m = 5A + 5B

l = 10A + 5B + 5C

 

Then,

s = 2(m + l). Substituting gives the third plane:

5A + 10B + 5C = 2(5A + 5B + 10A + 5B + 5C), or 0 = 25A + 10B + 5C