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Home » Let \(x\) be the width of the path. - Crosslake

Let \(x\) be the width of the path. - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 9\left(\frac{256}{81}n^2\right) + 16n^2 = \frac{256}{9}n^2 + 16n^2 = \frac{256 + 144}{9}n^2 = \frac{400}{9}n^2 = 25.

📌 Thus $ n^2 = \frac{25 \cdot 9}{400} = \frac{225}{400} = \frac{9}{16} \implies n = \pm \frac{3}{4} $.

📌 If $ n = \frac{3}{4} $, $ m = -\frac{16}{9} \cdot \frac{3}{4} = -\frac{4}{3} $. If $ n = -\frac{3}{4} $, $ m = \frac{4}{3} $.

📌 Check $ \vec{OG} \cdot (\vec{OA} + \vec{OB}) = 0 $: Both cases satisfy. However, the problem likely expects positive $ n $ (growth direction). Thus $ (m, n) = \left(-\frac{4}{3}, \frac{3}{4}\right) $.

📌 \boxed{\left( -\dfrac{4}{3}, \dfrac{3}{4} \right)}

📌 Question: The maximum imaginary part of a root of $ z^5 + z^3 + z + 1 = 0 $ can be expressed as $ \sin \theta $. Find $ \theta $ in degrees.

📌 Solution: Factor the polynomial:

📌 z^5 + z^3 + z + 1 = z^3(z^2 + 1) + 1(z + 1).

🔥 Popular Posts

For $ t = 2 $: $ 4a + 2b + c = 1200 $ (2)
For $ t = 3 $: $ 9a + 3b + c = 2100 $ (3)
Subtract (1) from (2): $ 3a + b = 700 $ (4)
Subtract (2) from (3): $ 5a + b = 900 $ (5)
Subtract (4) from (5): $ 2a = 200 \Rightarrow a = 100 $.
The value of $ a $ is $ \boxed{100} $.
Question: Rationalize the denominator of $ \frac{\sqrt{7}}{\sqrt{7} -...
Solution: Multiply numerator and denominator by the conjugate $ \sqrt{...
\frac{\sqrt{7}(\sqrt{7} + \sqrt{3})}{(\sqrt{7} - \sqrt{3})(\sqrt{7} +...
The rationalized form is $ \boxed{\dfrac{7 + \sqrt{21}}{4}} $.

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