The volume of one cloud, in cubic kilometers, grows according to the expression 4^3x – 1, where x is time in hours. Another cloud grows according to the expression 8x. After how many hours will the two clouds have the same volume? What is their volume, in cubic kilometers?

x = ⅔ hr

V = 4 km³

Step-by-step explanation:

The volume of the first cloud is given as:

V1 = 4^(3x - 1)

The volume of the second cloud is given as:

V2 = 8^x

When the two clouds have the same volume, V1 = V2:

=> 4^(3x - 1) = 8^x

We can rewrite this as:

2^[2(3x - 1)] = 2^[3(x)]

=> 2^(6x - 2) = 2^(3x)

According to the law of indices, we can equate the powers because their bases are equal (2 is the base).

=> 6x - 2 = 3x

6x - 3x = 2

3x = 2

x = ⅔ hr

Hence, after 2/3 hr, their volumes will be equal.

To find the volume after 2/3 hr, we can use the equation of either cloud but let us use the equation for the second cloud for simplicity sake:

V2 = 8^(⅔)

V2 = 4 km³ or 4 cubic kilometers