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Home » V = \frac13 \pi (3)^2 (9) - Red Crowns

V = \frac13 \pi (3)^2 (9) - Red Crowns

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 \frac{1}{4} s^2 = 36 \quad \Rightarrow \quad s^2 = 144 \quad \Rightarrow \quad s = 12 \text{ cm}

📌 The new side length is:

📌 s' = 12 + 2 = 14 \text{ cm}

📌 A' = \frac{\sqrt{3}}{4} (14)^2 = \frac{\sqrt{3}}{4} \cdot 196 = 49\sqrt{3} \text{ cm}^2

📌 \Delta A = A' - A = 49\sqrt{3} - 36\sqrt{3} = 13\sqrt{3} \text{ cm}^2

📌 Thus, the area increases by \(\boxed{13\sqrt{3}}\) square centimeters.Question: Find the area enclosed by the graph of $ |x| + |y| = 4 $.

📌 Solution: The equation $ |x| + |y| = 4 $ describes a diamond (or rhombus) centered at the origin. To find the vertices, consider the four cases for absolute values:

📌 $ x \geq 0, y \geq 0 $: $ x + y = 4 $, intercepts at $ (4, 0) $, $ (0, 4) $

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e 0$ â OK.
$\sin(90^\circ) = 1$, $\cos(90^\circ) = 0$ â $0
e 1$ â Wait! But $\cos z = 0$, so only valid if $2\sin z - 1 = 0$, b...
e 0$. Contradiction?
Wait â at $z = 90^\circ$, $\cos z = 0$, so LHS: $\sin(180^\circ) = 0...
$\sin(2 \cdot 90^\circ) = \sin(180^\circ) = 0$, $\cos(90^\circ) = 0$ â...
$\sin(270^\circ) = -1$, $\cos(270^\circ) = 0$ â $-1 = 0$? No â Not...
So only solutions where $\cos z = 0$ and $\sin z = 1/2$, but $\cos z =...
e 1/2$.
Only valid are $z = 30^\circ$ and $z = 150^\circ$.

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