The function *P*(*t*) = 145*e*^{-0.092t} models a runner’s pulse, *P*(*t*), in beats per minute, *t* minutes after a race, where 0 ≤ *t* ≤15. Graph the function using a graphing utility. TRACE along the graph and determine after how many minutes the runner’s pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

1) The original function is: **P(t) = 145e-0.092t**

2) When solving for 70 beats per minute, you replace P(t) with the number 70:

**70 = 145e-0.092t**

3) Then, you divide each side of the equation by 145 to work towards isolating (t).

70/145 = 145e-0.092t/145

**0.482758621 = e-0.092t**

** **

4) Then, you use the definition of a logarithm to continue to solve for t.

-0.092t = ln (0.482758621)

**-0.092t = -0.7282385**

5) Next, you divide each side of the equation by -0.092 to solve for t.

-0.092t/-0.092 = -0.7282385/-0.092

**t = 7.91563587**

6) Rounding the value of (t) to the nearest tenth of a minute:

**t = 7.9 minutes or 8 minutes**

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