The total resistance ZT in a circuit is given by Z sub t equals quantity z sub 1 times z sub 2 end quantity over quantity z sub 1 plus z sub 2 end quantity comma where z1 = 70 + 40i and z2 = 50 − 35i. What is the total resistance in standard form? Round to three decimal places.17.394 − 4.487i40.607 − 5.442i40.748 − 5.461i41.061 − 5.461i

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CalypsoE60768 CalypsoE60768 · Answer 1 · 2022-10-28T09:48:07+02:00

From the information given,

ZT = (Z1Z2)/(Z1 + Z2)

Z1 = 70 + 40i

Z2 = 50 - 35i

Z1Z2 = (70 + 40i)(50 - 35i)

Z1Z2 = 3500 - 2450i + 2000i - 1400i^2

Recall, i^2 = - 1

Z1Z2 = 3500 - 2450i + 2000i - 1400(-1)

Z1Z2 = 3500 - 2450i + 2000i + 1400

Z1Z2 = 4900 - 450i

Z1 + Z2 = 70 + 40i + 50 - 55i = 70 + 50 + 40i - 35i

Z1 + Z2 = 120 + 5i

ZT = (4900 - 450i)/(120 + 5i)

We would rationalize the denominator by multiplying by (120 - 5i)/(120 - 5i). It becomes

ZT = (4900 - 450i)/(120 + 5i) x (120 - 5i)/(120 - 5i)

ZT = (4900 - 450i)(120 - 5i)/(120 + 5i)(120 - 5i)

(4900 - 450i)(120 - 5i) = 588000 - 24500i - 54000i + 2250i^2

(4900 - 450i)(120 - 5i) = 588000 - 24500i - 54000i + 2250(- 1)

(4900 - 450i)(120 - 5i) = 588000 - 24500i - 54000i - 2250

(4900 - 450i)(120 - 5i) = 585750 - 78500i

(120 + 5i)(120 - 5i) = 14400 - 600i + 600i - 25i^2

(120 + 5i)(120 - 5i) = 14400 - 25i(-1)

(120 + 5i)(120 - 5i) = 14400 + 25

(120 + 5i)(120 - 5i) = `14425

ZT = (585750 - 78500i)/14425)

ZT = 40.607 - 5.442i