Solve. Approximately one-half of all aluminum cans...

Solve. Approximately one-half of all aluminum cans distributed will be recycled each year. A beverage company distributes $250,000$ cans. The number still in use after $t$ years is given by the function $$ N(t)=250,000\left(\frac{1}{2}\right)^{t} $$ a) After how many years will $60,000$ cans still be in use? b) After what amount of time will only 1000 cans still be in use?

Solve.
Approximately one-half of all aluminum cans distributed will be recycled each year. A beverage company distributes $250,000$ cans. The number still in use after $t$ years is given by the function
$$
N(t)=250,000\left(\frac{1}{2}\right)^{t}
$$
a) After how many years will $60,000$ cans still be in use?
b) After what amount of time will only 1000 cans still be in use?

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Key Concepts

Half-Life

The concept of half-life is central to understanding exponential decay processes. It is the time required for a quantity to reduce to half its initial value. In many applications, including the problem at hand, the half-life is used to determine how long it takes for a system to decrease to a specific fraction of its original amount.

Using Logarithms to Solve Exponential Equations

When faced with an exponential equation where the variable appears in the exponent, logarithms are a key tool for isolating that variable. By taking the logarithm of both sides of the equation, one can apply the power rule of logarithms to bring down the exponent, thereby transforming the equation to a linear form which is much easier to solve. This method is fundamental in solving time-to-decay problems and other similar exponential equations.

Exponential Decay Functions

Exponential decay functions describe processes in which a quantity decreases by a constant factor over equal time intervals. This mathematical model is characterized by the quantity being multiplied by a fixed fraction – in this case, one-half – during each time period. Such functions are widely used in fields ranging from radioactive decay to finance and recycling, providing a way to predict how quickly a system will decline over time.

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