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Home » Combined speed: \( 60 + 80 = 140 \) mph - Aurero

Combined speed: \( 60 + 80 = 140 \) mph - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Factor out \( u \):

📌 \[ u(u^2 - 3u + 2) = 0 \]

📌 \[ u(u - 1)(u - 2) = 0 \]

📌 So the roots are \( u = 0, 1, 2 \), all distinct.

📌 Thus, the number of distinct real roots is \(\boxed{3}\).Question: Compute $\tan 30^\circ$ in a right triangle with an opposite side of length $1$ and adjacent side of length $\sqrt{3}$.

📌 Solution: By definition, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$. Substituting the given values, $\tan 30^\circ = \frac{1}{\sqrt{3}}$. Rationalizing the denominator, $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$. Thus, $\boxed{\dfrac{\sqrt{3}}{3}}$.

📌 Question: A sundial's gnomon casts a shadow of length $5\,\text{cm}$ when the sun's elevation angle is $60^\circ$. What is the height of the gnomon?

📌 Solution: Using $\tan(60^\circ) = \frac{\text{height}}{\text{shadow length}}$, we substitute $\tan(60^\circ) = \sqrt{3}$ and shadow length $5\,\text{cm}$. Solving for height: $\text{height} = 5 \cdot \sqrt{3}$. Thus, $\boxed{5\sqrt{3}}$.

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