This problem models reservoir drainage where outflow rate follows Torricelli's law, making flow rate proportional to the square root of height. Since "outflow rate is proportional to √h(t)" and the reservoir is decreasing, we have dh/dt = -k√h where k > 0. The negative sign indicates decreasing water depth as water flows out. Choice B has the wrong sign representing filling rather than draining. Choice C would be simple exponential decay. Torricelli's law for fluid flow under gravity creates square root dependencies between height and flow rate in drainage problems.

This problem models treatment effectiveness where the rate of virus reduction depends on both current virus count and treatment dosage. The virus count "decreases due to treatment at a rate proportional to V(t) and to dosage d," giving dV/dt = -kdV where k > 0. The negative sign indicates decreasing virus count, and the rate depends on both the current viral load and the constant dosage strength. Choice B has an incorrect form with V and d in different relationships. Medical treatment models often show effectiveness proportional to both pathogen load and treatment intensity.

This problem models constant acceleration in physics. A "constant rate of 3 m/s²" means the velocity's rate of change is constant, so dv/dt = 3. This represents uniform acceleration where velocity increases by 3 m/s every second. Choice B (dv/dt = 3v) would model exponential growth of velocity, not constant acceleration. When the rate of change is described as constant, the differential equation has a constant on the right side, not a variable-dependent term.

This problem models logistic growth of bacteria with carrying capacity. The phrase "proportional to B and to (5000-B)" means dB/dt = kB(5000-B), combining growth proportional to current population with limitation from carrying capacity. This creates the characteristic S-shaped logistic curve where growth slows as B approaches 5000. Choice D (dB/dt = k(5000-B)) omits the B factor, missing the population-dependent growth aspect. For logistic growth, multiply the growth factor by both the population and the remaining capacity.

This problem requires modeling population growth with a differential equation. The phrase "grows at a rate proportional to its size" translates directly to dP/dt = kP, where k is the proportionality constant. The rate of change (dP/dt) equals k times the current population P. Choice C (dP/dt = k/P) incorrectly suggests growth slows as population increases, which contradicts proportional growth. When you see "rate proportional to [quantity]," write d[quantity]/dt = k × [that same quantity].

This problem involves modeling temperature change using Newton's Law of Cooling. The phrase "rate proportional to T-20" means dT/dt = k(T-20), where k is the proportionality constant. Since T represents the hot object's temperature and 20 is presumably room temperature, (T-20) represents the temperature difference. Choice B reverses the difference to (20-T), which would incorrectly suggest the object heats up when T > 20. For cooling/heating problems, always write the rate as proportional to (current temperature - ambient temperature).

This problem models rumor spread in a population using logistic dynamics. The rate "proportional to R(1-R)" translates directly to dR/dt = kR(1-R), where R represents the fraction who know the rumor. The R factor represents those who can spread it, while (1-R) represents those who can still learn it. Choice B (dR/dt = k(1-R)) misses the R factor, failing to account for the spreading population. For interaction models, multiply factors representing both interacting groups.

This problem models compound interest on a savings account. Interest at "4% per year, proportional to the current balance" translates to dS/dt = 0.04S, where 0.04 is the decimal form of 4%. The rate of change equals 0.04 times the current balance S. Choice A (dS/dt = 0.04) would add a constant $0.04 per year regardless of balance, which isn't how compound interest works. For percentage growth problems, convert the percentage to decimal and multiply by the current amount.

This problem combines natural population growth with constant immigration, requiring addition of two rate terms. The population changes due to both growth proportional to P(t) and constant immigration rate c, giving dP/dt = kP + c. The kP term represents natural population growth proportional to current population, while the +c term represents the constant rate of people moving into the city. Choice C incorrectly factors out k, changing the immigration term's effect. Choice E incorrectly makes immigration depend on time rather than being constant. When modeling population with both proportional growth and constant immigration, add the proportional term and constant term separately.

This problem requires setting up a differential equation from a rate description involving proportional decay. The tank leaks at a rate proportional to the current amount V(t), which means the rate of change of volume is negative and directly related to the volume itself. Since "proportional to the amount present" means the leak rate equals k times V(t), we get dV/dt = -kV where k > 0. The negative sign is crucial because the volume is decreasing due to leaking. Choice A represents constant rate change, not proportional to volume, while choice D has the wrong form with V in the denominator. Always ensure the sign reflects whether the quantity is increasing or decreasing, and check that proportional relationships translate to multiplication by a constant.