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A water tank has the shape of an inverte...

A water tank has the shape of an inverted righ circular cone with its axis vertical and vertex lowermost . Its semi-vertical angle is `tan^(-1)(0.5)` . Water is poured into it at a constant rate of 4 cubic meter per hour . Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 2 m.

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To solve the problem, we need to find the rate at which the level of water is rising in a conical tank when the depth of water is 2 meters. We will use the relationship between the volume of the cone and the height of the water. ### Step-by-Step Solution: 1. **Understand the Geometry of the Cone**: The tank is an inverted right circular cone with a semi-vertical angle given by \( \tan^{-1}(0.5) \). This means that if we let \( \alpha \) be the semi-vertical angle, then: \[ \tan(\alpha) = 0.5 \implies \frac{r}{h} = 0.5 \implies r = 0.5h ...
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