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Home » Substitute the values: \( V = 3.14 \times 5^2 \times 10 \). - Get link 4share

Substitute the values: \( V = 3.14 \times 5^2 \times 10 \). - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 \frac{10!}{4! \cdot 5! \cdot 1!}

📌 10! = 3628800,\quad 4! = 24,\quad 5! = 120,\quad 1! = 1

📌 \frac{3628800}{24 \cdot 120 \cdot 1} = \frac{3628800}{2880} = 1260

📌 Thus, the number of unique activation sequences is $\boxed{1260}$.

📌 \boxed{1260}

📌 Question: A startup uses a rotation-based irrigation system activated by 5 identical solar valves, 3 identical pressure regulators, and 2 identical flow meters. If each component is engaged once over 10 days, how many distinct activation orders exist?

📌 Solution: We seek the number of distinct permutations of a multiset: 10 components — 5 identical solar valves (V), 3 identical pressure regulators (P), and 2 identical flow meters (F). The total number of sequences is:

📌 \frac{10!}{5! \cdot 3! \cdot 2!}

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V_1 = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
New dimensions:
New radius: \( 0.8r \)
New height: \( 1.5 \times 2r = 3r \)
New volume:
V_2 = \pi (0.8r)^2 (3r) = \pi (0.64r^2)(3r) = 1.92\pi r^3
Ratio:
\frac{V_2}{V_1} = \frac{1.92\pi r^3}{2\pi r^3} = 0.96
Thus, the ratio of the new volume to the original volume is:
\boxed{0.96}

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