Use the information in exercise 5 and assume that α=0.1. Find the time at which 50 % of oil rese

step 1 Let's talk about this question. We are given that a model for how long our aluminum resources wi

Use the information in exercise 5 and assume that α=0.1....

Key Concepts

Exponential Decay

This concept involves a process in which a quantity decreases over time at a rate proportional to its current value, typically modeled by the equation Q(t) = Q? e^(–?t). It provides a framework for describing situations where resources diminish steadily, and it is fundamental to understanding how factors such as decay constants affect the time dynamics of the process.

Half-Life

The half-life is the period over which half of a given initial quantity is depleted. In the context of exponential decay, it is derived by setting the remaining quantity to 50% of the initial amount, leading to a mathematical expression t?/? = ln(2)/?. This concept is critical when evaluating the time scale over which significant changes occur in a decaying system.

Logarithmic Equations

Logarithmic equations are used to solve for time or other variables in exponential models. By applying logarithms to both sides of an equation such as Q(t)/Q? = e^(–?t), one can extract the variable trapped in the exponent (e.g., solving for t). This approach is essential for finding precise time values corresponding to specific percentage changes in the modeled quantity.

Transcript

00:01 Let's talk about this question.

00:02 We are given that a model for how long our aluminum resources will last given by this logarithmic equation.

00:11 Where is the percentage increase in consumption from the current levels of use and tease them time in years before the resources is depleted.

00:23 So first of we need to use a graphical utility to graph this particular function.

00:27 So let's go to dismos and let's just go call this ft because it's dependent on, let's call this fr rather, because it's dependent on rate, which is represented by r.

00:38 So that's the natural log of 2 ,500.

00:41 So that's natural log of 2 ,500 r plus 1 over, over lnr plus 1.

00:49 So over lnr plus 1.

00:53 So that's how it's going to look like.

00:55 That would be the answer to our first question.

00:57 We need to grab this using the graphical utility and clearly that's how the curve is looking so uh let's talk about part b part b talks about if the consumption of the aluminum increases by five percentage per year five percentage per year then in how many years will deplete our aluminum resources so we are given that the value of the rate the rate is already given to us rate is given as five percentage and we need to find that in how many years will it will replenish.

01:31 In short, we need to find the functional value at this particular point.

01:37 And since this is the rate, so the rate must be taken and the decimal approximation is that's 0 .05.

01:43 So we need to find the value of f of 0 .05 because that is the rate at which we are consuming, we are consuming or the rate at which it is getting increased per year.

01:56 So it will last for 142 years...

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