Step 4 dA Since we need to find dt , we need an equation that relates the rates for the output variables. In dA dL dw other words, we need an equation that relates dt dt ., and dt To obtain the required related rates equation, use implicit differentiation to differentiate the equation from Step 3 with respect to t. Related rates equation: Step 5 dA To find dt we need to substitute values for each of the other quantities in the related rates dA dL dw dL dw equation W+ dt dt dt L. In Step 2 we identified the values for and dt dt But we also need to identify the values for L and W. Reread the problem context and the top of this page note the value of L and the value for W. L = ftStep 6 We are now ready to find the rate at which the area of the triangle is increasing when the length is 12 feet and the width is 6 feet. Gather the values you identified in Step 2 and Step 5 and substitute these values into the related dA dL dW rates e uation, : W L. q dt dt + dt How fast is the area of the rectangle increasing when the length is 12 feet and the width is 6 feet? dA , ft2 dt 3 Step 7 Finally, we write our result from Step 6 in context. The area of the triangle is increasing at a rate of . Let's Learn by Doing t The length of a rectangle is increasing at a rate of 2f. 5 it And the rectangle's width is increasing at a rate of 3f. 3 We'll use the step-by-step process below to answer the following question. How fast is the area of the rectangle increasing when - the length is 12 feet, and - the width is 6 feet? Step 2 Next we identify the known rates and the rate we need to find. For each of the following, if the rate is known, enter its value. If we do not yet know the rate, enter DNE. (Although we know that the DNE is only temporary, as the unknown rate will exist when we find it.) dL 7 ft dt s dW _ ft dt 5 CIA 7 #2 dt 3 Step 3 Now we're ready to write a formula that relates the output variables. Equation relating A, L, and W: C]