Select the correct answer from each drop-down menu.
Consider the following equation:
[tex]3^x=2^{-x}+4[/tex]
If the equation is solved using successive approximation, select the value of [tex]x[/tex] after the following number of iterations.
1 iteration: [tex]x=[/tex]
2 iteration: [tex]x=[/tex]
3 iteration: [tex]x=[/tex]
[tex]5 / 4[/tex]
[tex]7 / 4[/tex]
[tex]11 / 8[/tex]
[tex]3 / 2[/tex]
Answer (2)
Rewrite the equation as x = lo g 3 ( 2 − x + 4 ) .
Perform the first iteration: x 1 = lo g 3 ( 2 − 1 + 4 ) ≈ 1.369 , which is closest to 11/8 .
Perform the second iteration: x 2 = lo g 3 ( 2 − 11/8 + 4 ) ≈ 1.346 , which is closest to 11/8 .
Perform the third iteration: x 3 = lo g 3 ( 2 − 11/8 + 4 ) ≈ 1.347 , which is closest to 11/8 .
The approximate value of x after 1, 2, and 3 iterations is 8 11 .
Explanation
Understanding the Problem We are given the equation 3 x = 2 − x + 4 and asked to find the approximate values of x after 1, 2, and 3 iterations using successive approximation. The possible values for the approximations are 5/4 , 7/4 , 11/8 , and 3/2 .
Rewriting the Equation We can rewrite the equation as x = lo g 3 ( 2 − x + 4 ) . We will use this form for successive approximation. Let's start with an initial guess of x 0 = 1 .
First Iteration Iteration 1: Substitute x 0 = 1 into the equation to get x 1 = lo g 3 ( 2 − 1 + 4 ) = lo g 3 ( 4.5 ) = lo g 3 ( 9/2 ) . Using a calculator, we find that x 1 ≈ 1.369 . Among the given options, 11/8 = 1.375 is the closest to 1.369 .
Second Iteration Iteration 2: Substitute x 1 = 1.369 (or 11/8 = 1.375 ) into the equation to get x 2 = lo g 3 ( 2 − 1.369 + 4 ) ≈ lo g 3 ( 0.386 + 4 ) = lo g 3 ( 4.386 ) ≈ 1.346 . Among the given options, 11/8 = 1.375 is the closest to 1.346 .
Third Iteration Iteration 3: Substitute x 2 = 1.346 (or 11/8 = 1.375 ) into the equation to get x 3 = lo g 3 ( 2 − 1.346 + 4 ) ≈ lo g 3 ( 0.392 + 4 ) = lo g 3 ( 4.392 ) ≈ 1.347 . Among the given options, 11/8 = 1.375 is the closest to 1.347 .
Final Answer Therefore, after 1 iteration, x ≈ 11/8 . After 2 iterations, x ≈ 11/8 . After 3 iterations, x ≈ 11/8 .
Examples
Successive approximation is a method used to find approximate solutions to equations, especially when direct algebraic solutions are difficult or impossible to obtain. This technique is widely applied in engineering and physics to model and analyze systems where variables are interdependent. For example, in electrical engineering, it can be used to determine the current and voltage in complex circuits. In physics, it can be used to model the behavior of particles in a force field. The equation 3 x = 2 − x + 4 can be seen as a simplified model representing a balance between exponential growth ( 3 x ) and decay ( 2 − x ), offset by a constant value (4).
After performing successive approximations for the equation 3 x = 2 − x + 4 , the values of x after 1, 2, and 3 iterations are all closest to 8 11 . The chosen multiple choice option is 8 11 .
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