A model for how long our coal resources will last is given by T=(ln(300 r+1))/(ln(r+1)) where r

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

A model for how long our coal resources will last is given...

A model for how long our coal resources will last is given by $$T=\frac{\ln (300 r+1)}{\ln (r+1)}$$ where $r$ is the percent increase in consumption from current levels of use and $T$ is the time (in years) before the resource is depleted. a. Use a graphing utility to graph this equation. b. If our consumption of coal increases by $3 \%$ per year, in how many years will we deplete our coal resources? c. What percent increase in consumption of coal will deplete the resource in 100 years? Round to the nearest tenth of a percent.

A model for how long our coal resources will last is given by
$$T=\frac{\ln (300 r+1)}{\ln (r+1)}$$
where $r$ is the percent increase in consumption from current levels of use and $T$ is the time (in years) before the resource is depleted.
a. Use a graphing utility to graph this equation.
b. If our consumption of coal increases by $3 \%$ per year, in how many years will we deplete our coal resources?
c. What percent increase in consumption of coal will deplete the resource in 100 years? Round to the nearest tenth of a percent.

Key Concepts

Solving Logarithmic Equations

Solving logarithmic equations entails using the properties of logarithms to isolate and determine the values of unknown variables. This process is fundamental when dealing with equations that emerge from models involving exponential growth, enabling the extraction of meaningful time or rate relationships.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and serve as crucial tools for solving equations where the unknown variable is in an exponent. They simplify multiplicative processes and help in converting exponential growth or decay models into linear relationships that are easier to analyze.

Exponential Growth

Exponential growth occurs when the rate of change of a quantity is proportional to the current amount, leading to rapid increases over time. This concept is widely applied to model real-world phenomena like population dynamics, radioactive decay, and resource consumption, where small changes compound over time.

Mathematical Modeling

Mathematical modeling involves creating equations or expressions that represent real-world processes. Models are essential for predicting future behavior, making informed decisions, and understanding the interplay between various factors in scenarios such as resource depletion or growth phenomena.

Graphing Techniques

Graphing techniques using graphing utilities provide a visual representation of mathematical models, allowing one to inspect the behavior of functions, identify trends, and verify theoretical predictions. These techniques aid in understanding complex relationships and in communicating mathematical insights effectively.

Transcript

00:01 Let's talk about this question.

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

00:11 Where r is the percent increase is the consumption from current levels of use and tease the time in years before the resource is depleted.

00:20 We have to use a graphical utility to graph this particular equation.

00:24 So let's do that.

00:26 We have natural log of...

00:30 Natural log of 300 r plus 1 300 r plus 1 and over natural log of over natural log of r plus 1 and this can be represented as y is equal to so that's the required logarithmic curve which we are getting so part a is done part b says that if our resource concept increases by 3 % per year and 3 % per year in how many years will we deplete our core resources so in short we are given that the r which is the percentage increase in the consumption is given as 3 so we need to find a value of t when r is equal to 3 so that's what we have to do so in short we need to find the value if we need to find the value of t3 so that's going to be natural log of 300 times 3 plus 1 over natural log of 3 plus 1 this can be easily found graphically because let's call this f r and we need to find a value of f3 t3 or f3 is one in the same so if we find the value of f3 that's coming as 4 point uh it's coming as 4 .9 so that would be the, that would be the, that would be the, that would be the answer to this particular one.

02:09 There is one correction over here.

02:11 When we have to put the r, so that would be the decimal equivalent of this percentage.

02:17 So three percentage is nothing but 0 .03.

02:20 So we're going to replace r by 0 .03 and not 3.

02:24 So we've got to find the value of 0 .0 .0...

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