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Answer :
To find out how long it takes for the skydiver to reach [tex]\(90\%\)[/tex] of the terminal velocity, we need to use the given velocity formula:
[tex]\[ v = 186 \left(1 - 0.834^t\right) \][/tex]
1. Identify the terminal velocity and the target velocity:
- The terminal velocity, [tex]\(v_t\)[/tex], is [tex]\(186\)[/tex] feet per second.
- We want to find the time when the skydiver reaches [tex]\(90\%\)[/tex] of this terminal velocity:
[tex]\[ 90\% \text{ of } 186 = 0.90 \times 186 = 167.4 \text{ feet per second} \][/tex]
2. Set up the equation:
[tex]\[ 167.4 = 186\left(1 - 0.834^t\right) \][/tex]
3. Solve for [tex]\(t\)[/tex]:
- First, divide both sides by [tex]\(186\)[/tex]:
[tex]\[ \frac{167.4}{186} = 1 - 0.834^t \][/tex]
- Simplify the fraction:
[tex]\[ 0.90 = 1 - 0.834^t \][/tex]
- Rearrange to isolate the exponential term:
[tex]\[ 0.834^t = 1 - 0.90 \][/tex]
[tex]\[ 0.834^t = 0.10 \][/tex]
4. Take the logarithm of both sides to solve for [tex]\(t\)[/tex]:
[tex]\[ \log(0.834^t) = \log(0.10) \][/tex]
- Use the property of logarithms [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex]:
[tex]\[ t \cdot \log(0.834) = \log(0.10) \][/tex]
- Solve for [tex]\(t\)[/tex]:
[tex]\[ t = \frac{\log(0.10)}{\log(0.834)} \][/tex]
5. Calculate the result:
[tex]\[ t \approx 12.68 \text{ seconds} \][/tex]
So, it takes approximately [tex]\(12.68\)[/tex] seconds for the skydiver to reach [tex]\(90\%\)[/tex] of the terminal velocity.
[tex]\[ v = 186 \left(1 - 0.834^t\right) \][/tex]
1. Identify the terminal velocity and the target velocity:
- The terminal velocity, [tex]\(v_t\)[/tex], is [tex]\(186\)[/tex] feet per second.
- We want to find the time when the skydiver reaches [tex]\(90\%\)[/tex] of this terminal velocity:
[tex]\[ 90\% \text{ of } 186 = 0.90 \times 186 = 167.4 \text{ feet per second} \][/tex]
2. Set up the equation:
[tex]\[ 167.4 = 186\left(1 - 0.834^t\right) \][/tex]
3. Solve for [tex]\(t\)[/tex]:
- First, divide both sides by [tex]\(186\)[/tex]:
[tex]\[ \frac{167.4}{186} = 1 - 0.834^t \][/tex]
- Simplify the fraction:
[tex]\[ 0.90 = 1 - 0.834^t \][/tex]
- Rearrange to isolate the exponential term:
[tex]\[ 0.834^t = 1 - 0.90 \][/tex]
[tex]\[ 0.834^t = 0.10 \][/tex]
4. Take the logarithm of both sides to solve for [tex]\(t\)[/tex]:
[tex]\[ \log(0.834^t) = \log(0.10) \][/tex]
- Use the property of logarithms [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex]:
[tex]\[ t \cdot \log(0.834) = \log(0.10) \][/tex]
- Solve for [tex]\(t\)[/tex]:
[tex]\[ t = \frac{\log(0.10)}{\log(0.834)} \][/tex]
5. Calculate the result:
[tex]\[ t \approx 12.68 \text{ seconds} \][/tex]
So, it takes approximately [tex]\(12.68\)[/tex] seconds for the skydiver to reach [tex]\(90\%\)[/tex] of the terminal velocity.
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