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The speed, in miles per hour, of a parti...

The speed, in miles per hour, of a particular experimental spacecraft t minutes after it is lauched is modeled by the function M, which is defined as `M(t)=200(3)^((t)/(3))`. According to this model, what is the speed, in miles per hour, 9 minutes after the spacecraft is lauched?

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To find the speed of the spacecraft 9 minutes after it is launched, we will use the given function \( M(t) = 200 \cdot 3^{\frac{t}{3}} \). ### Step-by-Step Solution: 1. **Identify the function**: The speed of the spacecraft is modeled by the function \( M(t) = 200 \cdot 3^{\frac{t}{3}} \). 2. **Substitute the value of \( t \)**: We need to find the speed at \( t = 9 \) minutes. So, we will substitute \( t = 9 \) into the function: \[ M(9) = 200 \cdot 3^{\frac{9}{3}} \] 3. **Simplify the exponent**: Calculate \( \frac{9}{3} \): \[ \frac{9}{3} = 3 \] Therefore, we can rewrite the equation as: \[ M(9) = 200 \cdot 3^{3} \] 4. **Calculate \( 3^{3} \)**: Now, we need to calculate \( 3^{3} \): \[ 3^{3} = 27 \] 5. **Multiply by 200**: Now substitute back into the equation: \[ M(9) = 200 \cdot 27 \] 6. **Perform the multiplication**: Calculate \( 200 \cdot 27 \): \[ 200 \cdot 27 = 5400 \] 7. **Conclusion**: Thus, the speed of the spacecraft 9 minutes after it is launched is: \[ M(9) = 5400 \text{ miles per hour} \] ### Final Answer: The speed of the spacecraft 9 minutes after launch is **5400 miles per hour**.
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