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Home » Thus, the smallest such $n$ is $\boxed9$. - Tacotoon

Thus, the smallest such $n$ is $\boxed9$. - Tacotoon

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 A rectangle has a length that is 3 times its width. If the perimeter is 64 meters, what are the dimensions of the rectangle?

📌 The perimeter \( P \) is given by \( P = 2(\text{length} + \text{width}) = 2(3w + w) = 8w \).

📌 Setting \( 8w = 64 \), solve for \( w \): \( w = 8 \).

📌 The length is \( 3w = 24 \).

📌 The dimensions are 8 meters by 24 meters.

📌 A geometric series has a first term of 5 and a common ratio of 2. What is the sum of the first 6 terms?

📌 where \( a = 5 \), \( r = 2 \), and \( n = 6 \):

📌 S_6 = 5 \frac{2^6 - 1}{2 - 1} = 5 \times (64 - 1) = 5 \times 63 = 315

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Actually, the **maximum finite value of $r$** occurs when $\sin(2\thet...
r^2 \leq \frac{100}{1} = 100 \Rightarrow r \leq 10.
But we must check if this maximum is **attainable**. That is, does the...
Set $\sin(2\theta) = 1 \Rightarrow 2\theta = \frac{\pi}{2} + 2k\pi \Ri...
Try $\theta = \frac{\pi}{4}$:
Then $r^2 = \frac{100}{1} = 100 \Rightarrow r = 10$.
Check if this point lies within the circular region:
The circle has radius 10, so $r = 10$ is on the boundary — valid.
Thus, the **maximum possible value of $r$** is:
\boxed{10}

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