Mathematics
The number of contaminating particles on a silicon wafer prior to a certain rinsing process was determined for each wafer in a sample of size 100, resulting in the following frequencies:Number of particles: 0, 1, 2, 3, 4, , 5, 6, 7Frequency: 1, 2, 3, 12, 11, 15, 18, 10Number of particles: 8, 9, 10, 11, 12, 13, 14Frequency: 12, 4, 5, 3, 1, 2, 1(a.) What proportion of the sampled wafers had at least one particle? At least five particles?(b.) What proportion of the sampled wafers had between five and ten particles, inclusive? Strictly between five and ten particles?(c.) Draw a histogram using relative frequency on the vertical axis. How would you describe the shape of the histogram.
You would like to make a nutritious meal of eggs, mixed vegetables and brown rice. The meal should provide at least 35 g of carbohydrates, at least 30 g of protein, and no more than 45 g of fat. One serving of eggs contains 2 g of carbohydrates, 18 g of protein, and 12 g of fat. A serving of mixed vegetables contains 14 g of carbohydrates, 15 g of protein, and 8 g of fat. A serving of rice contains 40 g of carbohydrates, 6 g of protein, and 1 g of fat. A serving of eggs costs $3.75, a serving of mixed vegetables costs $3.50, and a serving of rice costs $2. It is possible to order a partial serving, e.g. 0.75 servings of rice. Formulate a linear optimization model that could be used to determine the number of servings of eggs, mixed vegetables, and rice for your meal that meets the nutrition requirements at minimal cost.