Chapter 7.2, Problem 2GP

a.

To determine

To Find:

The equation to model the distribution each year of cans that are made from recycled aluminium.

Answer to Problem 2GP

The equation is $y\approx \left(3200000\right){\left(0.66622\right)}^x$ .

Explanation of Solution

Given information:

At the beginning, the manufacturer distributed $3.2$ million aluminium cans in 2005.

After 5 years, the manufacturer distributed $4,20,000$ cans made from recycled ones.

Then the exponential function is given by

  $\begin{array}{l}y=a{b}^x\\ 420000=3200000\cdot b^5\\ \frac{42}{320}=b^5\\ \sqrt[5]{\frac{21}{160}}=b\\ b\approx 0.66622\end{array}$

An equation that model the aluminium cans is $y\approx \left(3200000\right){\left(0.66622\right)}^x$ .

Therefore, the equation is $y\approx \left(3200000\right){\left(0.66622\right)}^x$ .

b.

To determine

To Find:

The number of aluminium cans recycled cans be expected in the year 2050.

Answer to Problem 2GP

The approximated value is $y\approx 0.28174$ .

Explanation of Solution

Given information:

At the beginning, the manufacturer distributed $3.2$ million aluminium cans in 2005.

After 5 years, the manufacturer distributed $4,20,000$ cans made from recycled ones.

From part (a) the equation is $y\approx \left(3200000\right){\left(0.66622\right)}^x$

Replacing x with $40$

  $\begin{array}{l}y\approx \left(3200000\right){\left(0.66622\right)}^x\\ y\approx 0.28174\end{array}$

Therefore, the approximated value is $y\approx 0.28174$ .