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question:A student organization at a university sold French fries on Frenger Mall during the school year. The table below shows their weekly sales, and they estimate that they lost 10% to 50% of potential sales due to running out of inventory. Each box sold for 50. The group needs a recommendation for an inventory level for the next fall to avoid such inconvenience. How would you determine the appropriate inventory level, and what factors would support your recommendation? Week | Box of Fries Sold --- | --- 1 | 11 2 | 12 3 | 13 4 | 15 5 | 18 6 | 17 7 | 14 8 | 16 9 | 20 10 | 25 11 | 30 12 | 25 13 | 30 14 | 25 15 | 20
answer:To determine the appropriate inventory level, we'll calculate the potential sales with a 10% and 50% increase based on the actual sales and then find the average of these three values. This approach accounts for the uncertainty in the estimated lost sales. Here are the calculations: Week | Box of Fries Sold | Sales with 10% Increase | Sales with 50% Increase --- | --- | --- | --- 1 | 11 | 12.1 | 16.5 2 | 12 | 13.2 | 18.0 3 | 13 | 14.3 | 19.5 4 | 15 | 16.5 | 22.5 5 | 18 | 19.8 | 27.0 6 | 17 | 18.7 | 25.5 7 | 14 | 15.4 | 21.0 8 | 16 | 17.6 | 24.0 9 | 20 | 22.0 | 30.0 10 | 25 | 27.5 | 37.5 11 | 30 | 33.0 | 45.0 12 | 25 | 27.5 | 37.5 13 | 30 | 33.0 | 45.0 14 | 25 | 27.5 | 37.5 15 | 20 | 22.0 | 30.0 Average Sales: 19.8 (Actual), 21.9 (10% Increase), 27.2 (50% Increase) To minimize the risk of running out of inventory, we recommend stocking 27 boxes (rounded to the nearest whole number). This recommendation takes into account the average potential sales with a 50% increase, which is the higher end of the estimated lost sales range, ensuring that the organization is well-prepared for peak demand while minimizing the risk of stockouts.
question:Find the Jacobian matrix of the vector-valued function vec{F}(x, y, z) = leftlangle f(x, y, z), g(x, y, z), h(x, y, z) rightrangle, where f(x, y, z) = frac{1}{z^{3/2}}, g(x, y, z) = arcsin(y), and h(x, y, z) = arctan(z).
answer:The Jacobian matrix of vec{F}(x, y, z) is given by: J(vec{F}) = begin{bmatrix} frac{partial f}{partial x} & frac{partial f}{partial y} & frac{partial f}{partial z} frac{partial g}{partial x} & frac{partial g}{partial y} & frac{partial g}{partial z} frac{partial h}{partial x} & frac{partial h}{partial y} & frac{partial h}{partial z} end{bmatrix} = begin{bmatrix} 0 & 0 & -frac{3}{2 z^{5/2}} 0 & frac{1}{sqrt{1-y^2}} & 0 0 & 0 & frac{1}{z^2+1} end{bmatrix}
question:Solve the following system of two equations: 18x - 16y + 9 = 0, 5x - y + 20 = 0
answer:To solve the system of equations, we can use the substitution method. First, we solve the second equation for y: 5x - y + 20 = 0 -y = -5x - 20 y = 5x + 20 Now we can substitute this expression for y into the first equation: 18x - 16(5x + 20) + 9 = 0 18x - 80x - 320 + 9 = 0 -62x - 311 = 0 -62x = 311 x = -frac{311}{62} Now we can substitute this value of x back into the expression we found for y: y = 5x + 20 y = 5(-frac{311}{62}) + 20 y = -frac{315}{62} Therefore, the solution to the system of equations is x = -frac{311}{62} and y = -frac{315}{62}. The solution to the given system of equations is x = -frac{311}{62} and y = -frac{315}{62}.
question:Explain how market failures related to monopoly, externalities, and public goods impact the efficient allocation of resources in the energy sector.
answer:* Monopoly: Monopolies in the energy sector can prevent competition and lead to higher prices and reduced innovation. This can result in consumers paying more for energy than they would in a competitive market, and it can stifle the development of new and more efficient energy sources. * Externalities: Burning fossil fuels and other energy sources can create negative externalities, such as air pollution and climate change. These externalities are not reflected in the market price of energy, which can lead to overconsumption and environmental damage. * Public Goods: Some energy sources, such as solar and wind power, are public goods. This means that they are non-excludable and non-rivalrous, so it is difficult to charge consumers for their use. As a result, there may be underinvestment in these renewable energy sources, leading to a shortage of clean and affordable energy. These market failures can prevent the efficient allocation of resources in the energy sector, leading to higher costs, environmental damage, and a lack of access to clean and affordable energy.