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Solving for \(w\), we get \(w = 12\).
The length is \(2 \times 12 = 24\) meters.
The area is \(12 \times 24 = 288\) square meters.
The sum of the first \(n\) terms of an arithmetic sequence is given by \(S_n = 3n^2 + 5n\). Find the 10th term of the sequence.
The \(n\)-th term \(a_n\) is given by \(a_n = S_n - S_{n-1}\).
\(S_{10} = 3(10)^2 + 5(10) = 300 + 50 = 350\).
\(S_9 = 3(9)^2 + 5(9) = 243 + 45 = 288\).
Thus, \(a_{10} = 350 - 288 = 62\).
A train travels 120 km at a speed of \(v\) km/h and then 180 km at \(v + 20\) km/h. If the total time for the journey is 5 hours, find \(v\).
The time for the first part is \(\frac{120}{v}\) hours.