This question can be solved by using dimensions of S.I Units.
(a)
The dimensions of A and B are [LT⁻³] and [LT⁻¹], respectively.
Following are some important dimensions of base S.I units:
Mass = [M]
Length = [L]
Time = [T]
Now, we will solve for the dimensions of the given equation:
[tex]x = At^3 + Bt[/tex]
Here, it is given that x has dimensions of length, while t has the dimension of time. Therefore, substituting known dimensions we have:
[tex][L] = A[T^3] + B[T][/tex]
Now, according to the properties of dimensions, when two dimensions are added, the dimension of each term must be equal to the dimension of the term on the other side of the equation. It means:
Dimension = Dimension + Dimension
Hence, from our equation, we get two equations, as follows:
[tex][L] = A[T^3]\ |\ [L] = B[T]\\A = [LT^{-2}]\ |\ B = [LT^{-1}][/tex]
(b)
The dimension of dx/dt is [LT⁻¹].
Now, the dimension can be simply calculated by the formula:
[tex]\frac{dx}{dt} = 3At^2+B\\\\\frac{dx}{dt} = [LT^{-2}][T]^2+[LT^{-1}]\\\\\frac{dx}{dt} =[LT^{-1}]+[LT^{-1}]\\\\\frac{dx}{dt} =[LT^{-1}][/tex]
The attached picture shows the dimensions of seven basic units.
Learn more about dimensions here:
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