What is the original value of an industrial machine that after six years is worth $7,500 given a decay rate of 0.25 per year?

To determine the original value of the industrial machine, we can use the formula for exponential decay, which is expressed as:

[ V = P(1 - r)^t ]

where:

• ( V ) is the value after time ( t ),

• ( P ) is the original value,

• ( r ) is the decay rate,

• ( t ) is the time in years.

In this problem, we know that the machine's value after six years (( V )) is $7,500, the decay rate (( r )) is 0.25 (or 25%), and the time (( t )) is 6 years. Therefore, we can rearrange the equation to solve for ( P ):

[ P = \frac{V}{(1 - r)^t} ]

Substituting the known values:

[ P = \frac{7500}{(1 - 0.25)^6} = \frac{7500}{(0.75)^6} ]

Calculating ( (0.75)^6 ) gives approximately 0.17803125. Now we can plug this back into our equation:

[ P = \frac{750