Analyzing related rates problems: expressions

the base B of T of a triangle is decreasing at a rate of 13 meters per hour and the height H of T of the triangle is increasing at a rate of 6 meters per hour at a certain instant T Sub Zero the base is 5 meters and the height is 1 meter what is the rate of change of the area a of T so our area is going to be a function of T what is the rate of change of the area of the triangle at that instant and so what we're doing this exercise instead of going straight and trying to solve it what we need to do here is to identify the various units of different expressions and then try to think about what information is given and what's not and then that will actually equip us to actually solve this rate of change problem so let's just do this first part let's match each expression with its units and like always pause the video and see if you can do it on your own alright so the first one is B prime of T so this is the rate of change at which of which the base is changing with respect to time so if we think about it B of T that is the base that is going to be in meters so this is going to be in meters if we say B prime of T this is going to be how much our base is changing with respect to time so this is going to be meters per and they give us right over here they say it's decreasing at a rate of 13 meters per hour so the unit's here are meters per hour and so B prime of T that is going to be in meters per hour a at time T sub 0 remember a is the area of our triangle and we're measuring everything in meters you can tell from the information they've given us and so area is going to be in square units and so it's going to be in square meters now the height at time T sub 0 well both the base and the height those are lengths they're going to be measured in meters and so our height time T Sub Zero is going to be in meters and then here we have the rate of change of our area with respect to time so our area we already know is in meter squared but we want to know this here the this is going to be the rate of change of our area with respect to time so it's going to be an amount of area per unit time and time here we're using hours as you can see from some of the information they've given us so this is going to be area per unit time or meter squared per hour so it's going to be right over here that's area per unit time and the length we're using in this is meters and time is hours all right now they say match each expression with its given value so what is the base of the triangle at time T sub zero do they give that to us well let's see they say at a certain time at a certain instant T Sub Zero the base I'm gonna underline this in a different color the base at a certain instant T Sub Zero the base is five meters so they say the base at time T Sub Zero the base is a function of time but they tell us that it is five meters so this is five meters right over here now what about the rate of change of the base with respect to time do they tell us that well look right over here yeah that's actually the first piece of information they gave us the base B of T of a triangle is decreasing at a rate of 13 meters per hour so the rate of change of the base that is B prime of T which is equal to DB DT and they tell us that that is it's decreasing at a rate of 13 meters per hour so that would be negative 13 meters per hour and so the rate of change of the base with respect to time is going to be negative 13 they gave us that now a prime of T this is the rate of change of the area at time T sub zero did they give us this well they asked us that what is the rate of change of the area ace a of T of the triangle at that instant so this is what we actually need to figure out but they haven't given it to us otherwise there's no problem to solve so this one right over here is not given in fact this is what we are trying to solve for and then finally we have the chain the first derivative of the height with respect to time so you could view this as d H DT what is this going to be do they give it to us well look right over here they say the height of the triangle is increasing at a rate of 6 meters per hour so if they're saying H of T is increasing they're telling us the rate of change of H of T with respect to time so that's H prime of T and they're telling us that it is increasing at 6 meters per hour so it's going to be positive 6 meters per hour so they did indeed give us that now why is all of this a useful exercise to go through well now we are really ready to solve the question because in general if we're talking about any triangle we know that area is equal to 1/2 base times height now in this situation area and our base and our height they're all going to be functions of T so we could write a of T is equal to 1/2 times B of T times H of T and if we want to find the rate of change of our area at that instant and the instant that they're talking about is at time T sub 0 well then what we would want to do is take the derivative of both sides with respect to T so the derivative on the left hand side with respect to T would be a prime of T and then on the right hand side it would be 1 half times and we would actually use a combination of well it's really just the product rule right over here the derivative of the first function with respect to T so it's B prime of T times the second function this is just the product rule here plus the first function B of T times the derivative of the second function with respect to time and we need to figure out not just the general expression they want us to know what the rate of change of the area so a prime of T at that instant at T Sub Zero so what we want to figure out we want to figure out a prime of at time T Sub Zero well that's just going to be equal to one half times B prime of T Sub Zero times H of T Sub Zero plus B of T Sub Zero times H prime of T Sub Zero now that this might seem daunting except they've given us a lot of this information what is B prime of T Sub Zero well they tell us the rate of change of B with respect to time and it seems like it's going just gonna stay at negative 13 meters per hour so they gave us this H what is the height at time T Sub Zero well they tell us right over here at a certain instant the base is 5 meters and the height is 1 meter so they give us both B and H at T Sub Zero so they gave us this they gave us this and what is the rate of change of the height at time T Sub Zero well they tell us the height of the triangle is increasing at a rate of 6 meters per hour so they tell us that as well all of that stuff is given and so you just have to plug it in to figure out what is the rate of change of the area at T sub zero at that instant