If x varies inversely as $$y^3 - 1$$​ and is equal to 8 when y = 3, find x when y = 6.

Correct option is D

Given:
x varies inversely as $$y^3 - 1 .$$​​
When y = 3 , x = 8 .
Find  x  when  y = 6 .
Formula Used:
Inverse variation is defined as:
x =$$\frac{k}{y^3 - 1}$$​​
where k  is the constant of variation.
Solution:
Use the given values  y = 3  and  x = 8  to find the constant k :
x = $$\frac{k}{y^3 - 1}$$​​
8 = $$\frac{k}{3^3 - 1}$$​​
8 =$$\frac{k}{27 - 1}$$​

​8 =$$\frac{k}{26}$$​​

k =$$8 \times 26$$​​
k = 208
Now, substitute k = 208 and  y = 6 into the formula to find x :
x = $$\frac{208}{6^3 - 1}$$​​
x =$$\frac{208}{216 - 1}$$​​

​$$x = \frac{208}{215}$$​​
Alternate Method:
Since  x  varies inversely as  $$y^3 - 1$$​, we can write:
​$$x_1 (y_1^3 - 1) = x_2 (y_2^3 - 1)$$​​
where $$x_1 = 8 , y_1 = 3$$​ , and $$y_2 = 6$$​​
​$$8 (3^3 - 1) = x_2 (6^3 - 1)$$​​
​$$8 (27 - 1) = x_2 (216 - 1)$$​​
​$$8 (26) = x_2 (215)$$​​
​$$208 = 215 x_2$$​​
​$$x_2 = \frac{208}{215}$$​​

​$$x = \frac{208}{215}$$

Option (d) is right.