54. The volume of certain rectangular solid is 12x. If the dimensions of the solid are the integers x,y,and z,what is the greratest possible value of z?
a. 36
b. 24
c. 12
d. 6
e. 4
please answer and explain
you know the volume is
x*y*z = 12x
So, y*z = 12
y cannot be less than 1, so z cannot be more than 12
To find the greatest possible value of z, we need to factorize the given expression for the volume, which is 12x.
The prime factorization of 12x is 2^2 * 3 * x.
To maximize the value of z, we want to maximize the number of 2's, 3's, and x's that we can get.
Since z is an integer, it can be any factor of 12x. The largest factor of 12x will be obtained by dividing 12x by the smallest prime factor, which is 2.
Dividing 12x by 2 gives us 2 * 3 * x = 6x.
Now we can see that the largest possible value of z is equal to 6.
Therefore, the correct answer is:
d. 6