To find the time [tex]\( t \)[/tex] when the sky diver's velocity reaches 43 meters per second, given the velocity function [tex]\( v(t) = 61 - 61 e^{-0.26 t} \)[/tex], we need to set [tex]\( v(t) \)[/tex] equal to 43 and solve for [tex]\( t \)[/tex].
Start with the equation:
[tex]\[
v(t) = 43
\][/tex]
Substitute the given velocity function into this equation:
[tex]\[
61 - 61 e^{-0.26 t} = 43
\][/tex]
Next, isolate the exponential term. Subtract 61 from both sides:
[tex]\[
43 - 61 = -61 e^{-0.26 t}
\][/tex]
[tex]\[
-18 = -61 e^{-0.26 t}
\][/tex]
Divide both sides by -61 to solve for the exponential term:
[tex]\[
\frac{-18}{-61} = e^{-0.26 t}
\][/tex]
[tex]\[
\frac{18}{61} = e^{-0.26 t}
\][/tex]
Now, take the natural logarithm of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[
\ln\left(\frac{18}{61}\right) = -0.26 t
\][/tex]
Solve for [tex]\( t \)[/tex] by dividing both sides of the equation by -0.26:
[tex]\[
t = \frac{\ln\left(\frac{18}{61}\right)}{-0.26}
\][/tex]
At this point, calculate the expression using the exact value inside the logarithm before dividing:
[tex]\[
t \approx \frac{\ln(0.29508)}{-0.26}
\][/tex]
This yields a numerical value:
[tex]\[
t \approx 4.694238870296718
\][/tex]
Round this value to the nearest tenth:
[tex]\[
t \approx 4.7
\][/tex]
Thus, the sky diver's velocity will be 43 meters per second approximately [tex]\( \boxed{4.7} \)[/tex] seconds after diving.
