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Maximising Revenue?
A company manufactures and sells x units of a product per day. The selling price per unit is ₹(100 - 2x) and the cost price per unit is ₹(60 + 0.5x). How many units should be produced for maximum revenue?
Electric Shielding: A Conundrum
A thin layer of conducting material, such as a sheet of copper, is placed between a point charge and a free charge. What effect will the shielding have on the free charge if it is (i) positive and (ii) negative, respectively?
Minimizing Production Costs?
A fertilizer factory produces two types of fertilizers, X and Y. The cost of production for X is ₹5 per unit, and for Y is ₹8 per unit. However, producing Y requires an additional machinery rental of ₹50 per week. If the factory aims to meet a weekly demand of 120 units of X and 150 units of Y, and the total weekly production budget is ₹16000, how should the production be planned to minimize costs?
City's Water Resource Management?
The city of Noida has a constant water inflow of 1000 cubic meters per hour. It has two reservoirs, A and B, with capacities of 2,00,000 cubic meters and 3,00,000 cubic meters respectively. Using definite integrals, determine the number of hours for which the reservoirs can supply water at a rate of 800 cubic meters per hour to the residents of Noida.
Magnetic Field of a Solenoid
A long solenoid with 200 turns per meter and a current of 5 A is placed near a compass. The compass needle shows deflection. Assuming the Earth's magnetic field is negligible, determine the direction of the magnetic field at the compass and the number of turns per meter required to produce a magnetic field of 0.5 T at the compass, given the current is doubled to 10 A.
Magnetic Field Around a Coil?
A current-carrying coil of 50 turns is placed in a horizontal plane. The current flowing through the coil is 2A. At a distance of 10cm from the coil's center, a magnetic field of 5 x 10^-6 T is measured using a magnetometer. How will the magnetic field strength change if the number of turns in the coil is doubled, while keeping the current and distance constant?
Tanking Capacity of a Tank?
A cylindrical water tank of height 10 m and radius 4 m is to be built. If the tank's capacity is to be maximized, what should be the shape of the tank, and how would you determine its optimal dimensions using an integral?
Optimizing Witness Cost?
The Mumbai Police Department wants to allocate its personnel for surveillance in a high-security area. There are two types of witnesses: those who can identify a suspect with high accuracy and those who can do so with moderate accuracy. A high-accuracy witness costs ₹30,000 per day, while a moderate-accuracy witness costs ₹15,000 per day. For a specific event, the Police Department has a budget of ₹1,50,000. If the probability of the suspect attending the event is 0.7, and the probability of a high-accuracy witness correctly identifying the suspect is 0.85, while that of a moderate-accuracy witness is 0.6, how many high-accuracy and moderate-accuracy witnesses should be hired to achieve the highest possible probability of identifying the suspect?
Electric Field due to Spherical Shell?
A spherical shell of radius 10 cm has a total charge of 4 μC uniformly distributed over its surface. If a tiny test charge of -2 μC is placed at the centre of the shell, will it experience any net electric field and what would be its direction?
Maximizing Profit with Resource Constraints?
A company manufacturing smartphones and laptops has 800 hours of labour and 400 units of memory available. The profit from each smartphone is ₹800 and from each laptop is ₹1200. Formulate a linear programming problem to maximize the profit, considering the constraints on labour and memory.
Cascading Capacitors: A Charging Conundrum?
Three capacitors of capacitances 2 μF, 3 μF, and 4 μF are connected in cascade. If a potential difference of 12 V is applied across the combination, determine the potential difference across each capacitor and the total charge stored in the circuit. Assume the capacitors are initially uncharged and the charging process is ideal.
Maximizing Profit with Constraints?
Anju's Bakery produces two types of cakes - chocolate and vanilla. The profit from each chocolate cake is ₹ 15 and from each vanilla cake is ₹ 10. However, production is limited by a 4-hour workday, where each chocolate cake requires 2 hours and each vanilla cake requires 1.5 hours. Formulate a linear programming model to determine the optimal number of chocolate and vanilla cakes to produce for maximum profit.
Inverse Trigonometric Functions: Problem of Real-World Significance?
The inverse trigonometric functions are essential in calculating the angle of elevation for a ladder placed against a building. A 20-foot ladder is placed against a building with its foot 5 feet away from the wall. If the top of the ladder is 15 feet above the ground, can you determine the angle of elevation using inverse trigonometric functions?
Induction in a Coil?
A coil of insulated copper wire is connected to an ammeter and a galvanometer in series. When a magnet is brought near the coil, the galvanometer shows a deflection, but the ammeter does not. Explain the reason behind this observation, and suggest a method to improve the sensitivity of the galvanometer reading.
Solving a Real-World Problem with Inverse Trigonometric Functions?
A projectile is launched from the ground at an angle of 60 degrees above the horizontal with an initial velocity of 20 m/s. Assuming negligible air resistance, find the maximum height reached by the projectile if it lands 50 meters away from the point of launch. Use inverse trigonometric functions to solve this problem.
Maximizing Revenue: An E-commerce Dilemma?
A small e-commerce business sells a product at an initial price of $100. Due to market competition, the demand for the product is decreasing at a rate of 2% per day. To maximize revenue, the business needs to adjust the product price. Using the demand function, find the optimal price at which the revenue is maximized.
Traffic Modelling Conundrum?
The city traffic engineer observes that the rate at which people arrive at a popular bus stop is given by the function f(t) = 2t^2 + 3t + 1, where t is the time in hours. Assuming the bus arrives once every 4 hours, calculate the average number of people waiting at the bus stop, excluding the last group that arrives right before the bus, over a period of 12 hours. Assume the rate of people leaving the bus stop is negligible.
Capacitance in Parallel: A Surprising Twist?
When two capacitors of different capacitances are connected in parallel, does the total capacitance always increase? Consider a circuit where one capacitor has a capacitance of 10 μF and the other has a capacitance of 20 μF. What happens to the total capacitance when they are connected in parallel?
Critical Points on a Curve?
Let f(x) = |x^2 - 4x| + 1. Show that the function is differentiable at all points except at x = 2 and x = 0, and justify your answer with proper reasoning.
Interpreting Velocity Vectors in Projectile Motion?
A particle is projected from ground level with an initial velocity of 25 m/s at an angle of 60° to the horizontal. Assuming the acceleration due to gravity is 9.8 m/s² downward, represent the velocity of the particle 2 seconds after projection in terms of its horizontal and vertical components, and analyze its effect on the trajectory of the particle.
Falling Object with Air Resistance?
A stone is dropped from a height of 100 m above the ground where a strong wind is blowing upwards with a speed of 2 m/s. Using an appropriate function to model its descent, find the distance fallen by the stone in 3 seconds.
Inverse Trigonometry in Real-Life Scenario?
A company uses inverse trigonometric functions to model the angle of a solar panel to maximize energy absorption. If the panel's angle is inversely proportional to the cosine of the angle of elevation, and the initial angle is 60°, how would you adjust the angle of the solar panel to absorb the maximum amount of sunlight during a day when the angle of elevation is 30°?
Matrix Transpose and Determinant?
If A is a 2x2 matrix and AT represents the transpose of matrix A, then what can you infer about the relationship between the determinant of A and the determinant of AT when A is a singular matrix?
Optimization in Resource Allocation?
A small town is planning to build a new hospital, and the town council has decided to allocate the available funds between two projects: a new wing for the existing hospital and a new healthcare center for the underprivileged population. The town council has allocated a total of ₹5 crores for this project. The new hospital wing requires an initial investment of ₹2 crores and an ongoing maintenance cost of ₹0.8 lakhs per year. The new healthcare center requires an initial investment of ₹1 crore and an ongoing maintenance cost of ₹0.5 lakhs per year. What is the optimal allocation of funds between the two projects to maximize the utilization of the available funds?
Two Insulators in Contact?
When a glass rod and an ebonite rod are brought into contact and separated again, an ebonite rod is found to be negatively charged. Explain the process that occurs during contact and the resulting charge distribution on the two rods.
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