m3 A cylindrical tank has a radius of 5 m. Water is pumped into the tank at a rate of 4 . How fast is mm the height of the water increasing? WARNING: As you work through each step, keep in mind that math is case sensitive. So, if a quantity is represented by an uppercase letter or a lowercase letter, you will need to use the same case when typing the letter. Also, when you enter a function or an equation in an answer textbox, click on the Vars tab of the textbox to learn which letters are available and which case you should use. Step 1 . dV m3 . . . . . . Since we have E : 4 _ , and Since we need to find how fast the height of the water is increasmg, mm dh , we need a formula relating V and h. dt The volume of any right circular cylinder is V : wrgh. So, the volume of the water in the cylindrical tank is given by V : 7T?"2h where r is the radius of the tank and h, is the height of the water. Let's identify our input and output variables. , . dV m3 The given rate is : 4 , dt mm input variable is measured in minutes. , so we know the output variables are measured in meters, and the Output variables: V : volume of the water in cubic meters h : height of the water in meters Input variable: t : time in minutes Wait! The volume of the water is V : Tl'Tgh. So, did we forget to list if\" as an output variable? No! Since r = 5 m is a fixed value, r does not vary. In other words, r is a constant, not a variable. And, since r is a constant, we can substitute r = 5 m into V = Tr-h now (before we differentiate the equation). V = 25Th Step 2 Since we need to find dh dt we need an equation that relates the rates for the dV dh output variables. In other words, we need an equation that relates dt and dt To obtain the required related rates equation, use implicit differentiation to differentiate V = 257th with respect to t. Related rates equation: Step 3 dV Substitute the given value for dt - into the equation from Step 2 and solve for dh dt dh m dt (no rounding) S Step 4 Finally, we state the solution in context. The height of the water is increasing at a rate of m SStep 2 dh Since we need to find , we need an equation that relates the rates for the alt dV dh output variables. In other words, we need an equation that relates E and E. To obtain the required related rates equation, use implicit differentiation to differentiate V : 257% with respect to t. dV d]? - . : 9!\" _ Related rates equation. of: - )iI dr 0" Step 3 . . dV . . Substitute the given value for E into the equation from Step 2 and solve for dh dt ' dh _ 1 m d' E 7 3751: x ? (no roun ing) Step 4 Finally, we state the solution in context. The height of the water is increasing at a rate of