About three-fifths of all aluminum cans distributed will be...

About three-fifths of all aluminum cans distributed will be recycled each year. A beverage company distributes $250,000$ cans. The number still in use after time $t,$ in years, is given by the exponential function $$ N(t)=250,000\left(\frac{3}{3}\right)^{t} $$ a) The aluminum from how many cans is still in use after 0 year? 1 year? 4 years? 10 years? b) Graph the function.

About three-fifths of all aluminum cans distributed will be recycled each year. A beverage company distributes $250,000$ cans. The number still in use after time $t,$ in years, is given by the exponential function
$$
N(t)=250,000\left(\frac{3}{3}\right)^{t}
$$
a) The aluminum from how many cans is still in use after 0 year? 1 year? 4 years? 10 years?
b) Graph the function.

Key Concepts

Graphing Exponential Functions

Graphing an exponential function involves plotting points corresponding to various t values and connecting them to reveal the curve’s characteristic shape: a rapid increase or decrease with a horizontal asymptote, typically the x-axis. The graph visually demonstrates how the initial value and the decay/growth factor affect the behavior of the function over time.

Decay/Growth Factor

The decay or growth factor, often denoted by b in the function N(t) = N? * b?, determines whether the function models exponential growth (b > 1) or decay (0 < b < 1). In contexts such as recycling, this factor represents the proportion of the material that remains in use over each time period.

Function Evaluation

Evaluating an exponential function at specific time values involves substituting these values for t in the equation and computing the result. This approach is essential for predicting outcomes or determining the quantity remaining after a specified number of time units.

Exponential Functions

An exponential function is one of the form N(t) = N? * b?, where N? is the initial quantity, b is the constant base or rate factor, and t represents time. This type of function is used to model situations where a quantity grows or decays at a rate proportional to its current value.

Initial Value

The initial value represents the amount present at the start (time t = 0) of the process, and in exponential functions it is given by N?. This value serves as the reference point for the entire function and determines the position of its graph on the vertical axis.

Transcript

Please give Ace some feedback