the value of each set of coins varies as the square of its diameter, if its thickness remains constant, and it varies as the thickness, if the diameter remains constant. if the diameter of two coins are in the ratio 4:3, what should be the ratio of their thickness, be if the value of the first is four times that of the second?

We are given that the value of a coin varies with the square of its diameter and its thickness
Let us assume that the diameter of coin 1 is 4d units
Then the diameter of coin 2 is 3d units
Let us also assume that the thickness of coins 1 and 2 are x units and y units respectively
The price of a coin varies as the square of its diameter if its thickness remains constant, and it varies as the thickness if the diameter remains constant.
Hence, the price of a coin will be=k(diameter*diameter)*thickness
Where k is a constant value
Now, price of coin 1 is = k*4d*4d*x=16k(d*d)*x
Then , price of coin 2 is = k*3d*3d*y=9k(d*d)*y
Also, the value of the first is four times that of the second
16k(d*d)*x=4*9k(d*d)*y
x/y=9/4
or, x:y=9:4
Hence, the ratio of the thickness of both coins is= 9:4
We can take the thickness of both coins in 9:4 and see that coin 1 will be 4 times the value of coin 2