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(a) Assume the equation x = At3 + Bt describes the motion of a particular object, with x having the dimension of length and t having the dimension of time. Determine the dimensions of the constants A and B. (Use the following as necessary: L and T, where L is the unit of length and T is the unit of time.)

[A] = ?

[B] = ?

(b) Determine the dimensions of the derivative

dx/dt = 3At2 + B.

(Use the following as necessary: L and T, where L is the unit of length and T is the unit of time.)

[dx/dt] =?

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(a)

X = A t^3 + B t

Dimensional formula of A t^3 = Dimensional formula of X

Dimensional formula of A = [L] / [T^3]

= [ L T -3 ]

Dimensional formula of B t = Dimensional formula of X

Dimensional formula of B = [L] / [T]

= [ L T -1 ]

(b)

dx/dt = 3 A t^2 + B

Dimensional formula of dx/dt = Dimensional formula of B

= [ L T -1 ]

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x = At3 + Bt

x =L t =T

L = AT3 + BT

LT-1 = AT2 + B

so B = LT-1

LT-1 = AT2

A = LT-3

so A = LT-3 and B = LT-1

dx/dt = 3At2 + B. =LT-3 (T2) + LT-1 = LT-1

dx/dt = LT-1 (velocity)

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