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Magnetic Field Around a Coil?
A coil of insulated copper wire carrying a current of 5 A is placed near a compass needle. What would be the direction of the magnetic field lines around the coil, and how would it affect the compass needle's orientation?
Rainfall Volume Calculation?
A dam's reservoir has a rectangular cross-section with a length of 100 m and a variable width, represented by the function w(x) = 5x + 2, where x is the distance from the left end in meters. If the water depth is 3 meters, calculate the volume of water in the reservoir between the points where x = 0 and x = 20 m.
Capacitor in a Non-Uniform Field?
A parallel plate capacitor is placed in a non-uniform electric field. If one plate is taken out of the field, how will the value of capacitance change, and what effect will it have on the stored charge?
Population Growth Model?
The rate of change of a population's size is proportional to the current population. Using this information, derive a differential equation to model the population growth, assuming the initial population is 1000 and the growth rate constant is 0.02.
Modeling Population Growth
A certain species of fish grows at a rate proportional to its current population. If the initial population is 100 and the growth rate is 25% per year, use the differential equation to determine the population after 5 years. Consider the implications of this growth on the ecosystem.
Resistance in a Parallel Circuit?
In a parallel configuration of two resistors, one with a resistance of 10 Ω and the other with a resistance of 20 Ω, a voltage of 20 V is applied across each. How do the equivalent resistance and current through each resistor compare to their individual values?
Magnetic Field and Torque?
A bar magnet of length 10 cm is placed inside a solenoid of 100 turns per meter. If the solenoid carries a current of 2 A, what is the magnitude of the torque experienced by the bar magnet? Assume the magnetic moment of the bar magnet is 0.5 A·m².
Resistors in Parallel: A Conundrum
Three identical light bulbs, each with a resistance of 10 ohms, are connected in parallel across a 100-volt power source. If two of the bulbs are removed, how will the equivalent resistance of the remaining configuration change, and what will be the new current through the circuit?
Direction Ratios of Resultant Vector?
If A = 2i - 3j + 4k and B = -3i + 2j - 5k are two vectors, find the direction ratios of the resultant vector when A is multiplied by 2 and added to B.
Optimization of a Production Process?
A factory produces metal rods of uniform length and diameter. The cost of raw materials and production is directly proportional to the surface area of the rod. If the factory produces rods of length 1 meter, find the optimal diameter that minimizes the production cost. Assume the cost is given by C = k(2πrh), where k is a constant, r is the radius, and h is the height of the rod.
Modelling Population Growth?
The population of a certain city is growing at a rate proportional to the current population. If the initial population is 1,00,000 and the growth rate is 5% per year, use the appropriate definite integral to find the population after 10 years.
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