Problem 55 - Applied Calculus

Question

World consumption of lead is running at the rate of \(5 e^{0.05 t}\) million metric tons per year, where \(t\) is measured in years and \(t=0\) corresponds to 2014 a. Find a formula for the total amount of lead that will be consumed within \(t\) years of 2014 b. Use a graphing calculator to find when the world's known resources of 89 million metric tons of lead will be exhausted. [Hint: Use INTERSECT.] Lead has many uses, from batteries to shields against radioactivity.

Step-by-Step Solution

Answer

a. Formula: \( C(t) = 100(e^{0.05t} - 1) \); b. Resources exhausted by mid-2027.

Step 1: Understand the Consumption Rate Function

The rate of lead consumption is given by the function \( R(t) = 5e^{0.05t} \) million metric tons per year. Here, \( t = 0 \) corresponds to the year 2014. This function shows us how much lead is consumed each year.

Step 2: Set Up the Integral for Part (a)

To find the total amount of lead consumed over \( t \) years, we need to integrate the rate of consumption function \( R(t) = 5e^{0.05t} \) with respect to \( t \). The formula for total consumption \( C(t) \) is given by:\[C(t) = \int_0^t 5e^{0.05x} \, dx\]

Step 3: Solve the Integral

Calculate the integral:\[C(t) = \int_0^t 5e^{0.05x} \, dx = \left[ \frac{5}{0.05} e^{0.05x} \right]_0^t = 100\left(e^{0.05t} - 1\right)\]Thus, the formula for the total amount of lead consumed within \( t \) years is \( C(t) = 100(e^{0.05t} - 1) \) million metric tons.

Step 4: Set Up the Equation for Part (b)

We need to find when the total consumption \( C(t) \) reaches 89 million metric tons. Set up the equation:\[ 100(e^{0.05t} - 1) = 89 \]Solve for \( t \) to find when the world's known resources will be exhausted.

Step 5: Solve for \( t \) Numerically

Rearrange the equation:\[ e^{0.05t} - 1 = 0.89 \] \[ e^{0.05t} = 1.89 \]Taking the natural logarithm of both sides gives:\[ 0.05t = \ ln(1.89) \]\[ t = \ \frac{ln(1.89)}{0.05} \]Using a calculator, we find \( t \approx 13.186 \) years.

Step 6: Determine the Year Resources Will Be Exhausted

Since \( t = 0 \) corresponds to the year 2014, add \( 13.186 \) years to 2014 to find the year resources will be exhausted:\[ 2014 + 13.186 \approx 2027.186 \]Hence, the resources will be exhausted in approximately mid-2027.

Key Concepts

Exponential Functions

Exponential functions are powerful mathematical tools that model growth or decay processes. The function given in the exercise is an example: \( R(t) = 5e^{0.05t} \). This is an exponential function where the base \( e \) signifies growth. Here, the constant \( 0.05 \) is a growth rate. Each increase in \( t \) (years) results in the consumption rate increasing by the factor of \( e^{0.05} \). Exponential functions are used widely to denote rapid growth such as population growth, radioactive decay, and, in this case, lead consumption.

Integration

Integration helps us find the accumulation of quantities, such as total lead consumed over a period. In the context of this exercise, we integrate the rate function \( R(t) = 5e^{0.05t} \) to find total consumption \( C(t) \). The setup is:

• \(C(t) = \int_0^t 5e^{0.05x} \, dx\)

Solving this integral provides us with \(C(t) = 100(e^{0.05t} - 1)\). This result tells us the total lead consumed from the start year 2014 to any year \( t \). This process of integration is vital for converting rates into total amounts in real-life scenarios.

Numerical Methods

Numerical methods often provide solutions when analytical solutions are complex or impossible. Here, after calculating the integral, we need to find when total lead will reach 89 million metric tons. The equation becomes:

• \( 100(e^{0.05t} - 1) = 89 \)

Using algebra, rearrange to \( e^{0.05t} = 1.89 \) and solve for \( t \). Sometimes manual algebra isn't enough, and calculators or computer programs approximate the values using numerical methods. Such methods, including the use of natural logarithms here, are indispensable in solving real-world problems effectively.

Resource Depletion

Resource depletion refers to the consumption of a resource faster than it can be replenished. In this context, we look at the depletion of lead based on a consumption model. Calculating when depletion occurs helps in planning and management of resources. The exercise shows that at the constant growth rate, lead resources will be exhausted by mid-2027. Understanding resource depletion emphasizes the need for conservation, efficient usage, and exploring alternatives. Such calculations are central to sustainable development efforts.

Graphing Calculators

Graphing calculators are valuable tools for visualizing mathematical functions and solving equations. In this exercise, a graphing calculator helps determine precisely when the lead resources will reach full depletion by using the 'INTERSECT' feature. By graphing the function \( C(t) = 100(e^{0.05t} - 1) \) and the line \( y = 89 \), the intersection point gives the time \( t \). This matches with numerically solving for \( t \) approximately 13.186 years after 2014, resulting in mid-2027 as the depletion year. Such calculators simplify complex mathematical operations, proving useful in academic and scientific investigations.