Related rates

{"questions":[{"content":"If V = (4/3)πr³, what is dV/dt?<br>[[choice-rate-l1]]","widgets":{"choice-rate-l1":{"type":"choice","options":["4πr²·dr/dt","4πr²","(4/3)π·dr/dt","3r²·dr/dt"],"answer":[0],"explanations":["The chain rule gives dV/dt = 4πr²·dr/dt.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Differentiate with respect to r, then multiply by dr/dt."],"explanation":"The chain rule gives dV/dt = 4πr²·dr/dt."},{"content":"A square side grows at 2 cm/s. When the side is 4 cm, dA/dt equals [[input-rate-l2]]","widgets":{"input-rate-l2":{"type":"input","answer":"16"}},"hints":["A = s², so dA/dt = 2s·ds/dt."],"explanation":"2(4)(2) = 16 cm²/s."},{"content":"Water is draining from a tank. Which sign should dV/dt have?<br>[[choice-rate-l3]]","widgets":{"choice-rate-l3":{"type":"choice","options":["Negative","Positive","Zero in every case","No sign"],"answer":[0],"explanations":["A decreasing quantity has a negative time derivative.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["The volume is decreasing."],"explanation":"A decreasing quantity has a negative time derivative."},{"content":"When should the dimensions at a specific instant usually be substituted?<br>[[choice-rate-l4]]","widgets":{"choice-rate-l4":{"type":"choice","options":["After differentiating the relationship","Before defining variables","Before writing the model","Never"],"answer":[0],"explanations":["Substitute instant values into the rate equation, not the original model first.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Keep changing quantities variable during differentiation."],"explanation":"Substitute instant values into the rate equation, not the original model first."},{"content":"The units of dA/dt when length is measured in metres and time in seconds are<br>[[choice-rate-l5]]","widgets":{"choice-rate-l5":{"type":"choice","options":["m²/s","m/s","m³/s","s/m²"],"answer":[0],"explanations":["An area rate has area units per time.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Area is measured in square metres."],"explanation":"An area rate has area units per time."}],"mix":1}