Find the value of p so that the expression (15.97)³ + 1.4373 × p + (0.03)³ is a perfect cube.

Correct option is C

Given:

Expression: $$(15.97)^3 + 1.4373×p + (0.03)^3$$​​

Need value of (p) so the entire expression becomes a perfect cube

Concept Used:

Binomial identity for cubes:

​$$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$​​

Solution: 

Here, 15.97 = 16 - 0.03, so a = 16, b = 0.03

​$$(16 - 0.03)^3 + (0.03)^3 \\ \ \\ = 16^3 - 3(16)^2(0.03) + 3(16)(0.03)^2 \\ \ \\ = 4096 - 23.04 + 0.0432 \\ \ \\ = 4073.0032$$​​

Perfect cube condition: $$(15.97)^3 + (0.03)^3 + 1.4373p = 16^3$$​​

Now, 

4073.0032 + 1.4373p = 4096

1.4373p = 22.9968

p = $$\frac{22.9968}{1.4373}$$​ = 16