Numerical Methods For Engineers, 7 Ed
Numerical Methods For Engineers, 7 Ed
7th Edition
ISBN: 9789352602131
Author: Canale Chapra
Publisher: MCGRAW-HILL HIGHER EDUCATION
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Chapter 3, Problem 11P

Use 5-digit arithmetic with chopping to determine the roots of the following equation with Eqs. (3.12) and (3.13)

x 2 5000.002 x + 10

Compute percent relative errors for your results.

Expert Solution & Answer
Check Mark
To determine

To calculate: The roots of the equation x25000.002x+10 with 5-digit arithmeticchopping by the use of the formula x=b±b24ac2a and x=2cb±b24ac. Also, find the percent relative error.

Answer to Problem 11P

Solution:

The roots of the equation x25000.002x+10 by the use of the formula x=b±b24ac2a are 4999.9 with relative percent error of 0.002% and 0.05 with relative percent error of 2400%.

And, the roots of the equation x25000.002x+10 by the use of the formula x=2cb±b24ac are 200 with relative percent error of 96% and 0.002 with relative percent error of approximately 0%.

Explanation of Solution

Given:

The equation, x25000.002x+10.

Formula used:

The quadratic formula for the equation ax2+bx+c=0 is,

x=b±b24ac2a

Alternative formula for roots when b2>>4ac,

x=2cb±b24ac

Relative error formula:

Relative error = | true valueapproximate valuetrue value |

Calculation:

Consider the following equation,

x25000.002x+10=0

Here, a=1,b=5000.002 and c=10.

Therefore, the roots of the equation from the quadratic formula is,

x=5000.002±(5000.002)24×102=5000.002±(5000.002)24×102=5000.002±4999.9982

Thus,

x1=5000.002+4999.9982=100002=5000

And,

x2=5000.0024999.9982=0.0042=0.002

Now, chop to 5 digits and find the root as below,

x=5000.0±(5000.0)24×102=5000.0±250000004×102=5000.0±249999602=5000.0±4999.92

Solve for two different roots:

x1=5000.0+4999.92=9999.92=4999.9

And,

x2=5000.04999.92=0.12=0.05

Hence, the relative percent error of the first root 4999.9 is,

Error=|50004999.95000|×100%=|0.15000|×100%=0.002%

And, the relative percent error of the second root 0.05 is,

Error=|0.0020.050.002|×100%=|0.0480.002|×100%=2400%

Hence, the roots of the equations are 4999.9 with relative percent error of 0.002% and 0.05 with relative percent error of 2400%.

Now, for the provided equation b2>>4ac, Therefore, use alternative formula to find the roots by chopping to 5-digit,

2cb±b24ac=2(10)5000.0±(5000.0)24×10=2(10)5000.0±250000004×10=205000.0±24999960=205000.0±4999.9

Solve for two different roots,

x1=205000.+4999.9=200.1=200

And,

x1=205000.04999.9=209999.9=0.002

Therefore, the relative percent error of the first root 200 is,

 Error=|50002005000|×100%=|48005000|×100%=96%

And, the relative percent error of the second root 0.00200002 is,

Error=|0.0020.0020.002|×100%=|00.002|×100%=0%

Hence, the roots of the equations are 200 with relative percent error of 96% and 0.002 with relative percent error of 0%.

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