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The probability of choosing one from each category is: - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Solution: The diagonal of the rectangle is the circle's diameter. Using the Pythagorean theorem: $ \text{Diagonal} = \sqrt{3^2 + 4^2} = 5 $ cm. The circumference is $ \pi \cdot \text{diameter} = 5\pi $ cm. Thus, the circumference is $ \boxed{5\pi} $ cm.

📌 Question: In a climate simulation, a wind turbine's foundation is a semicircular drum with diameter 12 meters. What is the radius of the drum?

📌 Solution: The diameter of the semicircular drum is 12 meters, so the radius is half the diameter. Thus, $ r = \frac{12}{2} = 6 $ meters. The radius is $ \boxed{6} $ meters.Question: Find $ x $ such that the vectors $ \begin{pmatrix} x \\ 3 \end{pmatrix} $ and $ \begin{pmatrix} 2 \\ -x \end{pmatrix} $ are orthogonal.

📌 Solution: Two vectors are orthogonal if their dot product equals zero. Compute the dot product: $ x \cdot 2 + 3 \cdot (-x) = 2x - 3x = -x $. Set this equal to zero: $ -x = 0 $. Solving gives $ x = 0 $. \boxed{0}

📌 Question: Determine the minimum value of $ (\tan x + \cot x)^2 $ for $ x $ in the domain where $ \tan x $ and $ \cot x $ are defined.

📌 Solution: Expand $ (\tan x + \cot x)^2 = \tan^2 x + 2 \tan x \cot x + \cot^2 x $. Simplify using $ \tan x \cot x = 1 $: $ \tan^2 x + 2 + \cot^2 x $. Use identity $ \tan^2 x + \cot^2 x = (\tan x - \cot x)^2 + 2 $, so expression becomes $ (\tan x - \cot x)^2 + 4 $. The minimum occurs when $ \tan x = \cot x $, i.e., $ x = \frac{\pi}{4} + k\frac{\pi}{2} $, giving $ 0 + 4 = 4 $. \boxed{4}

📌 Question: Compute $ (\cos \theta + i \sin \theta)^5 $ using De Moivre’s theorem.

📌 Solution: By De Moivre’s theorem, $ (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) $. Applying $ n = 5 $, the result is $ \cos(5\theta) + i \sin(5\theta) $. \boxed{\cos(5\theta) + i \sin(5\theta)}

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