Question

A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate R (in liters per hours) at a time t (in hours) is given in the table.a and c only

Answer 1

A table is given. It is required to use the linear regression capabilities to find a linear model for (t, ln(R)) with an equation of the form ln(R)=at+b, and then write in exponential form.

It is also required to use a graphing utility to draw a graph.

(a) Rewrite the table values for t and ln(R) as follows:

Input the values into a graphing utility to find the linear model:

`\ln(R)=-0.6259t+6.0770`

Take the exponent of both sides of the equation to write in exponential form as required:

`\begin{gathered} \Rightarrow e^(\ln(R))=e^(-0.6259t+6.0770) \\ \text{ Using the powers of sum property:} \\ \Rightarrow R=e^(6.0770)\cdot e^(-0.6259t) \\ \Rightarrow R=435.7201(e^(-0.6259t)) \end{gathered}`

(b) Use the graphing utility to graph the exponential function:

Notice that the graph at the top-right best fits the graph.

(c) Evaluate the definite integral using the graphing utility:

`\int_0^4435.7201e^(-0.6259t)dt=639.2116`

The answer is 639.2116 L.