Final answer:
To find the two numbers, we use the definition of mean and third proportionals, ending up with x = 18 and y = 8. However, none of the provided answer choices match these numbers, indicating a possible typo in the question or choices.
Step-by-step explanation:
The student is asking to find two numbers such that the mean proportional between them is 18, and the third proportional to them is 144.
To solve this problem, we need to understand that the mean proportional of two numbers x and y is √(xy), and the third proportional means if x and y are in proportion with y and z, then x : y :: y : z (which can be written as x/y = y/z, where z is the third proportional). Here, we need to find two numbers x and y such that √(xy) = 18 (which implies xy = 18² = 324), and y/x = 144/y.
Substituting y = 144/x into xy = 324 gives us x(144/x) = 324, so x² = 324. This leads us to x = 18. Then y can be found by y = 144/x which gives us y = 144/18 = 8. The numbers are therefore 18 and 8, which is not one of the answer choices provided. Considering this, there may be a typo in the question or the answer choices.