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Home » \cos \phi = \sqrt{1 - rac\sin^2 hetan^2} - Dyverse

\cos \phi = \sqrt{1 - rac\sin^2 hetan^2} - Dyverse

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Question: A museum curator divides a $12 \times 12$ meter exhibition hall into identical non-overlapping rectangular display zones of integer side lengths. What is the smallest possible number of such rectangles needed to exactly cover the floor if each rectangle must have area at least 18 square meters?

📌 Solution: The total area of the hall is $12 \times 12 = 144$ square meters. Each rectangle has area at least 18, so the maximum number of rectangles is $ \left\lfloor \frac{144}{18} \right\rfloor = 8 $, but we want the *minimum* number, so we seek the largest rectangle area that divides 144 and is $ \geq 18 $, and check if tiling with such rectangles covers the square.

📌 Try largest possible area dividing 144: $144$ (1 rectangle), but $144 \geq 18$, so possible only if one rectangle covers the whole hall. But is a single $12 \times 12$ rectangle allowed? Yes — it has area 144 ≥ 18, and dimensions are integers. So it can be tiled with one rectangle.

📌 Thus, the minimum number is $1$, since the full hall can be covered by a single valid rectangle.

📌 Question: A palynologist finds that $ \frac{2}{5} $ of the pollen grains in a sample are from coniferous trees, $ \frac{1}{3} $ from deciduous trees, and the rest from grasses. What fraction of the sample consists of grass pollen?

📌 Solution: Let the total be 1. Add the fractions of coniferous and deciduous:

📌 \frac{2}{5} + \frac{1}{3} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}

📌 The fraction from grasses is:

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