Question

The exponential decay of 500 grams of a substance is given by the equation Q=500*2^(-t), where Q is the amount of th equation represents an equivalent rate of decay of this substance?

Answer 1

The equivalent rate of decay for the substance is
`\( (dQ)/(dt) = -500 \ln(2) * 2^(-t) \)`.

Step-by-step explanation:

The exponential decay equation
`\( Q = 500 * 2^(-t) \)` describes the amount of the substance, Q, at any given time, t. To find the equivalent rate of decay, we take the derivative of Q with respect to time,
`\( (dQ)/(dt) \)`. The derivative of
`\( 2^(-t) \) is \( -\ln(2) * 2^(-t) \)`. Multiplying this by the coefficient of 500 in the original equation gives the equivalent rate of decay:
`\( (dQ)/(dt) = -500 \ln(2) * 2^(-t) \)`.

The negative sign indicates decay, and
`\( \ln(2) \)`is the natural logarithm of 2. This value arises because the base of the exponential function is 2. The equivalent rate of decay can be expressed as
`\( (dQ)/(dt) = -500 \ln(2) * 2^(-t) \)`, representing the instantaneous rate at which the substance is decreasing at any specific moment. This rate is dynamic, changing as time progresses due to the nature of exponential decay. Therefore, the negative coefficient and the involvement of
`\( \ln(2) \)` in the expression are crucial for accurately representing the decay of the substance over time.