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Home » The work is given by the line integral: - RoadRUNNER Motorcycle Touring & Travel Magazine

The work is given by the line integral: - RoadRUNNER Motorcycle Touring & Travel Magazine

📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 The volume \(V\) of a cone with radius \(r\) and height \(h\) is given by the formula:

📌 V_{\text{cone}} = \frac{1}{3} \pi r^2 h

📌 Substituting \(r = 3y\) and \(h = 4y\), we get:

📌 V_{\text{cone}} = \frac{1}{3} \pi (3y)^2 (4y) = \frac{1}{3} \pi \cdot 9y^2 \cdot 4y = \frac{1}{3} \pi \cdot 36y^3 = 12\pi y^3

📌 The volume \(V\) of a sphere with radius \(r\) is given by:

📌 Substituting \(r = 2y\), we find:

📌 V_{\text{sphere}} = \frac{4}{3} \pi (2y)^3 = \frac{4}{3} \pi \cdot 8y^3 = \frac{32}{3} \pi y^3

📌 Now, compute the ratio of the volume of the cone to the volume of the sphere:

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Now find intersection with original line \( 3x - 4y + 12 = 0 \).
Substitute \( y = -\frac{4}{3}x + \frac{10}{3} \):
3x - 4\left(-\frac{4}{3}x + \frac{10}{3}\right) + 12 = 0
\Rightarrow 3x + \frac{16}{3}x - \frac{40}{3} + 12 = 0
Multiply through by 3:
9x + 16x - 40 + 36 = 0 \Rightarrow 25x - 4 = 0 \Rightarrow x = \frac{4...
Now find \( y \):
y = -\frac{4}{3} \cdot \frac{4}{25} + \frac{10}{3} = -\frac{16}{75} +...
So the closest point is \( \left( \frac{4}{25}, \frac{78}{25} \right)...
Alternatively, use the formula for projection. The distance from point...

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