Question

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.

Answer 1

Given

`\begin{gathered} Z_T=(Z_1Z_2)/(Z_1+Z_2) \\ Z_1=70+40i,Z_2=50-35i_{} \end{gathered}`

First, obtain Z_1*Z_2 as shown below

`\begin{gathered} Z_1Z_2=\mleft(70+40i\mright)\mleft(50-35i\mright)=3500+2000i-2450i+1400 \\ \Rightarrow Z_1Z_2=4900-450i=50(98-9i) \end{gathered}`

On the other hand,

`Z_1+Z_2=(70+40i)+(50-35i)=120+5i=5(24+i)`

Thus,

`\Rightarrow Z_T=(50(98-9i))/(5(24+i))=(10(98-9i))/((24+i))`

Finally, the division between complex numbers is given by the formula below

`(a+ib)/(c+id)=((ac+bd)+(bc-ad)i)/(c^2+d^2)`

Therefore, in our case,

`\begin{gathered} \Rightarrow Z_T=10((98\cdot24-9\cdot1)+(-9\cdot24-98\cdot1)i)/(24^2+1^2) \\ \Rightarrow Z_T=10((2343-314i))/(577) \\ _{} \end{gathered}`

Rounding to three decimal places,

`\Rightarrow Z_T=40.607-5.442i`

Thus, the answer is the second option (top to bottom)