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Home » The ball will travel 120 meters in 4 seconds. - Get link 4share

The ball will travel 120 meters in 4 seconds. - Get link 4share

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Question: A museum curator studying a 19th-century celestial sphere needs to determine the angle between two star positions modeled by vectors $ \begin{pmatrix} 2 \\ -1 \end{pmatrix} $ and $ \begin{pmatrix} x \\ 3 \end{pmatrix} $, which are orthogonal. Find $ x $.

📌 Solution: For vectors to be orthogonal, their dot product must be zero: $ 2x + (-1)(3) = 0 $. Solving $ 2x - 3 = 0 $ gives $ x = \frac{3}{2} $. The ordered pair is $ \boxed{\left( \frac{3}{2}, 3 \right)} $.

📌 Question: A climate policy analyst models seasonal temperature variations with the function $ f(x) = 5\cos\left(\frac{\pi}{6}x\right) + 10 $. Determine the range of $ f(x) $ as $ x $ varies over all real numbers.

📌 Solution: The cosine function oscillates between $ -1 $ and $ 1 $, so $ 5\cos\left(\frac{\pi}{6}x\right) $ ranges from $ -5 $ to $ 5 $. Adding 10 shifts the range to $ [5, 15] $. The range is $ \boxed{[5, 15]} $.

📌 Question: A historian of science examines a 17th-century manuscript describing trigonometric identities. Find $ \tan \left( \arccos \left( \frac{1}{\sqrt{2}} \right) \right) $, reflecting early geometric reasoning.

📌 Solution: Let $ \theta = \arccos\left(\frac{1}{\sqrt{2}}\right) $, so $ \cos \theta = \frac{1}{\sqrt{2}} $. Then $ \theta = 45^\circ $, and $ \tan \theta = 1 $. The answer is $ \boxed{1} $.

📌 Question: A climate scientist analyzes urban heat island effects using a function $ g(x) = \sqrt{4 - x^2} $. Find the range of $ g(x) $ for real $ x $ where the expression is defined.

📌 Solution: The expression under the square root must satisfy $ 4 - x^2 \geq 0 $, so $ x \in [-2, 2] $. The maximum value of $ \sqrt{4 - x^2} $ is $ 2 $ (at $ x = 0 $), and the minimum is $ 0 $ (at $ x = \pm 2 $). The range is $ \boxed{[0, 2]} $.<think>

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1^2 + 2^2 + 3^2 + \dots + 8^2 = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64
+ 4 = 5 \\
+ 9 = 14 \\
+ 16 = 30 \\
+ 25 = 55 \\
+ 36 = 91 \\
+ 49 = 140 \\
+ 64 = 204
الآن، نجد باقي قسمة 204 على 10:
\div 10 = 20 \text{ مع باقٍ } 4

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