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1)  R = 7.62 Ω

   I = 3.15 A

   I₁ = 2.4 A

   I₂ = 0.75 A

2)  Rₐ = 11.25 Ω

       = 2.13 A

    Rₓ = 48 Ω

    = 0.5 A   

Explanation:

1) Given,

The resistor, R₁ = 10 Ω

The resistor, R₂ = 32 Ω

The voltage, V = 24 v

The effective resistance,

                                R = R₁ R₂ / (R₁ + R₂)

                                    = 10 x 32 / (10 + 32)

                                    = 7.62 Ω

Hence, the effective resistance, R = 7.62 Ω

The total current

                         I = V / R

                           = 24 / 7.62

                           = 3.15 A

Hence, the total current is the circuit, I = 3.15 A

The current through the first resistor,

                           I₁ = V / R₁

                              = 24 / 10

                              = 2.4 A

Hence, the current through the resistor, I₁ = 2.4 A

The current through the resistor, R₂

                          I₂ = V / R₂

                              = 24 / 32

                              = 0.75 A

Hence, the current through the resistor, I₂ = 0.75 A

2) Given,

The resistor, R₁ = 18  Ω

The resistor, R₂ = 30 Ω

The effective resistance in parallel connection,

                            Rₐ = R₁ R₂ / (R₁ + R₂)

                                 = 18 x 30 / (18 + 30)

                                 = 11.25 Ω

Hence, the effective resistance in parallel connection,  Rₐ = 11.25 Ω

The total current in the parallel connection is given by the formula,

                                = V/R

                                 = 24 / 11.25

                                 = 2.13 A

Hence, the current through the circuit in parallel connection is, = 2.13 A

Serial connection,

                                Rₓ  = R₁ + R₂

                                      = 18 + 30

                                      = 48 Ω

Hence, the effective resistance in series connection, Rₓ = 48 Ω

Current in series connection

                                    = V/R

                                              = 24 / 48

                                              = 0.5 A

Hence, the current through the circuit in parallel connection,   = 0.5 A     

the force required to move the cart is divided by three or 33.33 n.