Chapter ISG, Problem 7.5.1P

a.

To determine

To find: and write an equation that models the amount of gas left in the container after x hours, assuming there were 300 cubic centimeter in the container before leak. Then use this equation to find the amount of gas left in the container after 11 hours. Round answer to the nearest tenth.

Answer to Problem 7.5.1P

The equation that models the amount of gas left in the container after x hours will be:

  $y=300e^{-0.1x}.$

The amount of gas left in the container after 11 hours $y_{11}\approx 99.9$ cubic centimeters.

Explanation of Solution

Given information: There is a leak in a container that holds a certain nontoxic gas .Each

hour, it loses 10% of its volume, assuming there were 300 cubic centimeter in the container before leak.

Formula Used:

Since gas is leaking continuously so use here continuous decay formula,

  $A(t)=a{e}^{-rt}$ where a is the initial value, r is the rate of decaying substance, after t time.

Since gas is leaking, its loses 10% so r =0.10, initial volume of the gas a =300 cubic centimeter and t=x hours.

So, the equation that models the amount of gas left in the container after x hours will be:

  $y=300e^{-0.1x}.$

For finding amount of gas left in the container after 11 hours, substitute x =11 in the above equation.

  $y_{11}=300e^{-0.1(11)}=300e^{-1.1}\approx 99.9$ cubic centimeters.

Calculation:

b.

To determine

To find: that the graph of this function should be a scatter plot instead of a continues curve and how this relates to the domain of the function.

Answer to Problem 7.5.1P

The graph should be scatter plotter for small domain values and for large domain values it seems as curve.

Explanation of Solution

Given information: The function that models the amount of gas left in the container after x hours will be:

  $y=300e^{-0.1x}.$

The graph of the function $y=300e^{-0.1x}.$ is given below

  

It is clear from the graph if take large value of x then graph show as a curve but for smaller x values the graph seems as scatter plotter not a continues curve.

So, the graph should be scatter plotter for small domain values and for large domain values it seems as curve.

Calculation:

c.

To determine

To explain: why the domain of this function is infinite because as x get large function approaches but never equals to zero.

Answer to Problem 7.5.1P

It is clear from the graph of the function after a specified large value of x ( x > 100) the graph is flat on x − axis where function value is zero.

Explanation of Solution

Given information: The function that models the amount of gas left in the container after x hours will be:

  $y=300e^{-0.1x}.$

The graph of the function $y=300e^{-0.1x}.$ is given below

  

It is clear from the graph if take large value of x then graph show as a curve but for smaller x values the graph seems as scatter plotter not a continues curve.

So, the graph should be scatter plotter for small domain values and for large domain values it seems as curve.

While the output of an exponential function is never zero, but it can be so close to zero that for all practical purposes can accept zero as the answer.)

But this reasoning may not fit the context of the function because it is clear from the graph of the function after a specified large value of x ( x > 100) the graph is flat on x − axis where function value is zero.