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A motorcycle cost $12,000 when it was purchased. The value of a motorcycle decreases by 6% each year. Find the rate of decay each month and select the correct answer below.

−0.005143%
−0.5143%
−0.005%
−0.5%

Respuesta :

W0lf93

If the rate of decay is 6% per year, that means the motorcycle retains 93% of its value every year. Because there are 12 months in a year, you can use the following equation: 
 x^12 = 0.94, 
 where x = the rate of decay each month, 12 = the number of months, and 0.94 = the retained value each year. Next, set a logarithmic function on each side as such: 
 LOG(x^12) = LOG(0.94) 
 When applying log functions, exponentials (like the 12 in the equation) are moved outside of the function like so: 
 LOG(x^12) = 12(LOG(x)) 
 Therefore, 
 12(LOG(x)) = LOG(0.94) = -0.0268721464 
 When you divide both sides by 12, the equation becomes 
 LOG(x) = -0.00223934553 
 Finally, remove the LOG from the left side by applying both sides of the each by 10^() as such: 
 10^(LOG(X)) = 10^(-0.00223934553)
 X = 0.994856987 
 Therefore, the motorcycle retains that 0.994856987 of its value every year. 
 1 - 0.994856987 = 0.005143 
 Expressed as a percentage, this value is 0.5143013%

The rate of decay each month is given by 0.5143% and this can be determined by forming the expression from the given data.

Given :

  • A motorcycle cost $12,000 when it was purchased.
  • The value of a motorcycle decreases by 6% each year.

There are 12 months in a year then the value motorcycle retains every year is:

[tex]x^{12}=0.94[/tex]

Now, take the log on both sides in the above equation.

[tex]\rm logx^{12}=log(0.94)[/tex]

12 log x = log(0.94)

Now, divide log(0.94) by 12 in the above expression.

[tex]\rm log x = \dfrac{log(0.94)}{12}[/tex]

log x = -0.00223934553

[tex]x = 10^{-0.00223934553}[/tex]

x = 0.994856987

Now, to determine the rate of decay:

= (1 - x)100

= (1 - 0.994856987)100

= (0.005143)100

= 0.5143%

Therefore, the correct option is B).

For more information, refer to the link given below:

https://brainly.com/question/9695463

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