Related rates Test

{"questions":[{"content":"A circle has r = 6 cm and dr/dt = 2 cm/s. Enter the coefficient of π in dA/dt. [[input-rate-q1]]","widgets":{"input-rate-q1":{"type":"input","answer":"24"}},"hints":["Use dA/dt = 2πr·dr/dt."],"explanation":"[M1] Differentiate A = πr². [A1] dA/dt = 24π cm²/s.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["M1","A1"],"follow_through":false}},{"content":"A sphere radius decreases. Which formula correctly links the volume rate?<br>[[choice-rate-q2]]","widgets":{"choice-rate-q2":{"type":"choice","options":["dV/dt = 4πr²·dr/dt","dV/dt = 4πr²","dV/dt = (4/3)πr²·dr/dt","dV/dt = 3r²"],"answer":[0],"explanations":["[M1] Apply the chain rule. [A1] 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 (4/3)πr³."],"explanation":"[M1] Apply the chain rule. [A1] dV/dt = 4πr²·dr/dt.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["M1","A1"],"follow_through":false}},{"content":"A 13 m ladder has x = 5 m and dx/dt = 0.6 m/s. Which height should be used at this instant?<br>[[choice-rate-q3]]","widgets":{"choice-rate-q3":{"type":"choice","options":["12 m","8 m","18 m","√13 m"],"answer":[0],"explanations":["[M1] y² = 169 − 25. [A1] y = 12 m.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Use x² + y² = 13²."],"explanation":"[M1] y² = 169 − 25. [A1] y = 12 m.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["M1","A1"],"follow_through":false}},{"content":"The ladder calculation gives dy/dt = −0.25 m/s. Which interpretation earns the reasoning statement?<br>[[choice-rate-q4]]","widgets":{"choice-rate-q4":{"type":"choice","options":["The top of the ladder moves down at 0.25 m/s","The ladder length decreases","The height equals −0.25 m","The foot moves toward the wall"],"answer":[0],"explanations":["[R1] Negative dy/dt means decreasing height, so the top moves downward at 0.25 m/s. [FT] Interpret the sign consistently if the preceding calculated rate is carried forward.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Interpret sign, quantity and units."],"explanation":"[R1] Negative dy/dt means decreasing height, so the top moves downward at 0.25 m/s. [FT] Interpret the sign consistently if the preceding calculated rate is carried forward.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["R1"],"follow_through":true}},{"content":"A numerical model gives a changing area A(t). What written setup should precede a GDC derivative?<br>[[choice-rate-q5]]","widgets":{"choice-rate-q5":{"type":"choice","options":["Define dA/dt at the requested time and state its units","Only copy the decimal display","List calculator button names","Replace A with a constant"],"answer":[0],"explanations":["[M1] State the derivative being evaluated. [A1] Give the value. [R1] Interpret its sign and units.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Communicate the mathematical operation."],"explanation":"[M1] State the derivative being evaluated. [A1] Give the value. [R1] Interpret its sign and units.","assessment":{"paper_mode":"paper_2","calculator_mode":"expected","gdc_skill":"numerical_derivative","marks":["M1","A1","R1"],"follow_through":false}}],"mix":1}