# Interactive video lesson plan for: unit 4 test application of derivatives #8 related rates Cone conical tank volume height

#### Activity overview:

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Good day students welcome to mathgotserved.com in this clip were going to be going over problem number eight of our you need for test on the application of derivatives in this clip were going to be focusing on related rates okay before we can see there is the question let's take a look at the strategy for solving related rates problems okay so the strategy that's follows first of all you want to create a well label diagram applicable or sometimes you might want to generate a formula for equation and then one of find variable so all given quantities quantities to be determining number two you write an equation involving the variables is rates of change either our given or are to be determined now what you want to note is in some problems just passed the one we doing today you might have a variable was rate of change is not provided in that case you have to substitute out that variable making use of some geometric tools all the formula that you learn in algebra and then import three using the chain rule implicitly differentiate both sides of the equation respect to T after back to the comes up algebra problem when you goes that for which is substitute into the resulted equation all know values for the variables in the rates of change and so for the required rate of change our right so we have our little guy here on the right by reference and we going to apply it to problem number eight so is three X water is been poured into a conical X with the base radius of 10 feet and a depth of 12 feet it water is the import into the tank at the to cubic feet per minute how fast is the water level rising when the water level is four feet tall okay so the first thing we're going to do is we're going to create a well label diagram okay so there goes our clinical ten is inverted so we have the base radius is 10 feet so the base radius is indicate that can be right and then the height of the tank is swell see part a depth is anything that also so this is 12 feet the depth of the tan right and then let's take an average rate What we going to be calculating how fast on the water level is rising okay so if we take an average rate at say that H right here six is right here so for the average rate that this will be the radius R and in this me that this we the height H right here okay alright now let's go ahead and take a look at the question to see the rates that are provided in the rates that are not provided okay so in order to determine what variable should stay in our equation or formula want to look at the rates that are given and the rates that are to be determining okay the consult our strategy guide right here says rates that are given and once to be determining political one that she's okay so you have a question says what is the import into the tank asked to cubic feet per minute so this is what is going to cubic feet that is all volume so this right here is the be dv/dt is given okay so we know what given now let's see the rate other rate that's provided how fast is the water level rising we looking at the height how fast is the rising this is H T in this is what we are to find alright so let's go ahead and write equation that relates the volume and the height remember the volume of the common the formula for the vulnerable call is given by writing our squared H divided by three okay right now what you notice here which variables are to stay in which variable is to go noted that our strategy guide variables with the rates should be substituted out so we do not know what the rate of change of the radius is simply differentiates will will have dr /dt if we allow are to remain in this equation so we need to substitute out our six dr /dt is not given okay so let's write it down so out our since dr /dt is not given okay so the question is how do we substitute out our well you look at this situation here we have similar triangles so can use our knowledge from geometry a similar triangles to express our in terms of H okay so let's go ahead and draw our similar troubles here again so we have to similar right triangles so we have a bigger one and then swallow one so we can set up window to travel the similar correspondence I the proportion right so this is our and this right here is ten entire length is twelve in this piece right here is H okay so our over H is equal to our first so tan over twelve which corresponds to H is isolate our you have our equals you can reduce this right divided by two top and bottom so we have five over six H is our so what are we going to do next we once express this formula in terms of only the variables with the rates that are given or to be determining namely be and H so the going to have be equals prime is set up our I'm going to have five over six H okay some basically so six using this our Valley

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