To solve the problem of finding the initial velocity of the skydiver and the velocity after 5 seconds, we'll analyze the given velocity function:
[tex]\[ v(t) = 56 \left(1 - e^{-0.24 t} \right) \][/tex]
### Step-by-Step Solution:
1. Initial Velocity:
The initial velocity is the velocity at [tex]\( t = 0 \)[/tex] seconds. We will substitute [tex]\( t = 0 \)[/tex] into the velocity function.
[tex]\[ v(0) = 56 \left(1 - e^{-0.24 \cdot 0}\right) \][/tex]
Simplifying inside the parentheses:
[tex]\[ e^{-0.24 \cdot 0} = e^0 = 1 \][/tex]
Therefore,
[tex]\[ v(0) = 56 \left(1 - 1\right) = 56 \cdot 0 = 0 \][/tex]
So, the initial velocity is:
[tex]\[ \boxed{0} \, \text{m/s} \][/tex]
2. Velocity after 5 Seconds:
Next, we want to find the velocity at [tex]\( t = 5 \)[/tex] seconds. We substitute [tex]\( t = 5 \)[/tex] into the velocity function.
[tex]\[ v(5) = 56 \left(1 - e^{-0.24 \cdot 5}\right) \][/tex]
Simplifying the exponent:
[tex]\[ -0.24 \cdot 5 = -1.2 \][/tex]
Thus,
[tex]\[ v(5) = 56 \left(1 - e^{-1.2} \right) \][/tex]
We use the approximate value for [tex]\( e^{-1.2} \)[/tex] which is around 0.3012.
Therefore,
[tex]\[ v(5) = 56 \left(1 - 0.3012\right) \][/tex]
[tex]\[ v(5) = 56 \cdot 0.6988 \][/tex]
Calculating this product:
[tex]\[ v(5) \approx 39.9296 \][/tex]
Rounding to the nearest whole number gives:
[tex]\[ \boxed{39} \, \text{m/s} \][/tex]
### Summary:
- The initial velocity is [tex]\( \boxed{0} \)[/tex] m/s.
- The velocity after 5 seconds is [tex]\( \boxed{39} \)[/tex] m/s.
