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Home » How to Master Every Math Topic with Pauls’ Online Notes in Just Minutes! - DNSFLEX

How to Master Every Math Topic with Pauls’ Online Notes in Just Minutes! - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 \|\vec{d}(t)\|^2 = (2 - 2t)^2 + (-1 - t)^2 + (5 + t)^2

📌 Add them up:

📌 \|\vec{d}(t)\|^2 = (4 - 8t + 4t^2) + (1 + 2t + t^2) + (25 + 10t + t^2) = 6t^2 + 4t + 30

📌 Take derivative and set to 0:

📌 \frac{d}{dt}(6t^2 + 4t + 30) = 12t + 4 = 0 \Rightarrow t = -\frac{1}{3}

📌 Substitute $t = -\frac{1}{3}$ into $\vec{r}(t)$:

📌 \vec{r}\left(-\frac{1}{3}\right) = \begin{pmatrix} 1 + 2(-\frac{1}{3}) \\ -\frac{1}{3} \\ -3 - (-\frac{1}{3}) \end{pmatrix} = \begin{pmatrix} 1 - \frac{2}{3} \\ -\frac{1}{3} \\ -3 + \frac{1}{3} \end{pmatrix} = \begin{pmatrix} \frac{1}{3} \\ -\frac{1}{3} \\ -\frac{8}{3} \end{pmatrix}

📌 Thus, the closest point is:

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المحيط هو \(2(w + 2w) = 6w = 50\)، إذن \(w = \frac{50}{6} \approx 8.33...
الطول هو \(2 \times 8.33 \approx 16.67\).
المساحة هي \(w \times \text{الطول} = 8.33 \times 16.67 \approx 138.89\...
#### 138.89
تقدم شركة خصمًا بنسبة 15٪ على منتج سعره الأصلي 200 دولار، تليه خصم إضا...
الخصم الأول: \(200 \times 0.15 = 30\)، إذن السعر الجديد هو \(200 - 30...
الخصم الثاني: \(170 \times 0.10 = 17\)، إذن السعر النهائي هو \(170 - 1...
حل نظام المعادلات: \(2x + 3y = 12\) و \(x - y = 2\).
حل المعادلة الثانية لإيجاد \(x\): \(x = y + 2\).
عوض في المعادلة الأولى: \(2(y + 2) + 3y = 12\).

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