Answer:
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
The clouds will have the same volume of 2.8 square kilometers after 0.35 hours.
The clouds will have the same volume when:
4^(3x - 1) = 8x
Where these two are the expressions for the volumes of each cloud.
Notice that in the left side we have an exponential equation while on the right side we have a linear one, so we can't solve this analytically in a direct way.
What we can do is graph both curves and see when they intersect, like in the graph below:
There we can see that the curves intersect at x = 0.35.
The volume of the clouds at that time will be:
y = 8*0.35 = 2.8 square kilometers.
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