Draining a tank Water drains from the conical...

Question

Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft³/min.

a. What is the relation between the variables h and r in the figure?

Accepted Answer

Welcome back, everyone. A liquid is added to a conical tank at a constant rate of 5 ft cubs per minute. Based on the figure, what is the relationship between the radius R and height H of the liquid? We're given 4 answer choices A says radius R equals H divided by 4. B says R equals 4 H. C R equals 2 H and D R equals H divided by 2. So for this problem, we want to analyze. Similar triangles. What we're going to do is simply look at the given conical tank, and we're going to label. One of the resultant triangles which has side lengths of 10 ft and 20 ft, right, so essentially we have a green triangle and the other triangle that we want to analyze is going to be the red one with sidelines of R and H. Well done. What we can tell is that those two triangles are similar triangles, and we're going to use the property for similar triangles. It says that if we take the ratio of the corresponding sides for similar triangles, this ratio must be constant, right? The reason why these are similar triangles is because they contain exactly the same. Angles we can see that based on the fact that they share the same angle of 90 degrees, right? We have the right angle, then they share another common angle. We don't know what it is, but basically at the base of the cone we have a common angle which leaves us with the third exactly same angle based on the fact that we already have two exactly same angles. So now let's apply the property which says that if we analyze the green triangle, then the ratio between the height of 20. 20 ft And The radius of 10 ft. This ratio must be equal to the height H divided by the radius R. As we can see, we're basically taking the ratio of the corresponding sides. On the left hand side, 20 divided by 10 gives us 2. On the right hand side, we have H divided by R. According to the context of the problem, we are solving for R, so if we multiply both sides by R, this gives us 2 R equals H, and then we can divide both sides by 2, which gives us R equals H divided by 2. That would be our final answer and as we can see, we don't need the given rate of the liquid which is added to a conical thing, right? We only needed the original expression of R in terms of age, which essentially corresponds to the answer choice D. Thank you for watching.

Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft³/min.

a. What is the relation between the variables h and r in the figure?

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Key Concepts

Volume of a Cone

Related Rates

Geometric Relationships

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