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Home » Thus, the ratio is $ \boxed{\frac\pi6} $. - ToelettAPP

Thus, the ratio is $ \boxed{\frac\pi6} $. - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Solution: The probability of selecting no quantum dots (all nanotubes) is $\frac{\binom{7}{3}}{\binom{12}{3}} = \frac{35}{220}$. The complement gives the probability of at least one quantum dot: $1 - \frac{35}{220} = \frac{185}{220} = \frac{37}{44}$. \boxed{\dfrac{37}{44}}

📌 Question: A venture capitalist evaluates 9 clean energy startups, 4 of which are solar-based. If 5 startups are chosen randomly, what is the probability that exactly 3 are solar-based?

📌 Solution: The total number of ways to select 5 startups is $\binom{9}{5}$. The favorable case is $\binom{4}{3} \cdot \binom{5}{2}$ (3 solar and 2 non-solar). Thus, the probability is $\frac{4 \cdot 10}{126} = \frac{40}{126} = \frac{20}{63}$. \boxed{\dfrac{20}{63}}### Question 1

📌 A rectangular garden has a length that is twice its width. If the width is 10 meters, what is the area of the garden?

📌 The length of the garden is \(2 \times 10 = 20\) meters.

📌 The area is calculated as \( \text{length} \times \text{width} = 20 \times 10 = 200 \) square meters.

📌 A car travels at a constant speed of 80 km/h for 2.5 hours. How far does the car travel?

📌 Distance is calculated using the formula: \(\text{speed} \times \text{time} = 80 \times 2.5 = 200\) kilometers.

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So decrease in height is \( 12 - 2\sqrt{30} \) meters.
However, the problem asks for how far the top slides down — which is t...
- 2\sqrt{30}
But this is not simplified numerically.
Alternatively, is there a better way?
But 2√30 is exact.
However, check if 5-12-13 is correct — yes.
- 2\sqrt{30} \approx 1.05 \, \text{m}, \text{ but exact form: } 12 - 2...
But the decrease is \( 12 - \sqrt{120} \), and since \( \sqrt{120} = 2...
But perhaps leave as:

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