Question

A) As t increases without boundf(t) will approach _______. (Answer may contain unknown constants)B) The vertical axis intercept of the graph of f is: (__, ___) (Answer may contain unknown constants) C) While driving, Mychal accidentally runs over a nail and it punctures one of his tires. While waiting for help, he models the tire pressure over time. Initially the pressure is 35 psi (pounds per square inch) and when he checks after 3 minutes, the pressure is 29 psi. Find the values of constants Aand Bthat model tire pressure over timeA = ______B = ______D) Using Mychal's function model in part (c), determine the time at which the tire reaches the atmospheric pressure of about 14.74 psi. t = ______

Answer 1

Given the function:

`f(t)=(A)/(t+B)`

(a)

If t increases without bound, and given that A and B are constants, then the denominator increases without bound too. This means that the function decreases towards 0, then:

`\lim _(t\to\infty)f(t)=0`

(b)

The y-intercept of the graph can be calculated for t = 0:

`f(0)=(A)/(0+B)=(A)/(B)`

The coordinates of the y-intercept are:

`(0,(A)/(B))`

(c)

If the initial pressure (t = 0) is 35 psi:

`\begin{gathered} f(0)=35 \\ (A)/(B)=35 \\ \Rightarrow A=35B \end{gathered}`

Now, if after 3 minutes the pressure is 29 psi:

`\begin{gathered} f(3)=29 \\ (35B)/(3+B)=29 \\ 35B=87+29B \\ 6B=87 \\ \Rightarrow B=14.5 \end{gathered}`

Then:

`\begin{gathered} A=35B=35\cdot14.5 \\ \Rightarrow A=507.5 \end{gathered}`

(d)

Finally, if f(t) = 14.74:

`\begin{gathered} (507.5)/(t+14.5)=14.74 \\ 507.5=14.74t+213.73 \\ 14.74t=293.77 \\ \therefore t=19.930\text{ minutes} \end{gathered}`