answer:We assign east to the positive x direction and north to the positive y direction so that the person's velocity vector relative to the water is {eq}vec v_p = left< 0,2 right> {/eq} and the river's velocity vector is {eq}vec v_r = left< 4,0 right> {/eq}. Since the person is in the river, with respect to the riverbank he/she is actually moving with a velocity that is the sum of these two velocities. {eq}begin{align*} vec v &= vec v_p+vec v_r &= left< 0,2 right> + left< 4,0 right> &= left< 4,2 right> end{align*} {/eq} Then the angle as measured counterclockwise from the x-axis is {eq}begin{align*} tan theta &= frac{2}4 theta &approx boldsymbol{ 26.6^circ } end{align*} {/eq} Note that this direction is North of East. The speed of the person with respect to the riverbank is the magnitude of our vector: {eq}begin{align*} | left< 4,2 right> | &= sqrt{ left( 4 right)^2 + left( 2 right)^2} &approx boldsymbol{ 4.472 text{m/s} } end{align*} {/eq}

answer:To determine the optimal ordering quantity and the minimum total annual inventory cost, we can use the Economic Order Quantity (EOQ) formula: EOQ = √(2AS / H) where: A = Annual demand = 190 pounds per day * 365 days per year = 69,350 pounds per year S = Setup cost = 176 per order H = Holding cost = 6.50 per pound per year Plugging in the values, we get: EOQ = √(2 * 69,350 * 176 / 6.50) = 1937.93 pounds Since we cannot order a fraction of a pound, we round up the EOQ to the nearest whole number, which is 1938 pounds. The total number of orders per year can be calculated as: Number of orders = Annual demand / EOQ = 69,350 pounds / 1938 pounds = 35.78 orders Since we cannot place a fraction of an order, we round up the number of orders to the nearest whole number, which is 36 orders. The average inventory level can be calculated as: Average inventory = EOQ / 2 = 1938 pounds / 2 = 969 pounds The minimum total annual inventory cost can be calculated as: Total cost = (Number of orders * Setup cost) + (Average inventory * Holding cost) = (36 orders * 176 per order) + (969 pounds * 6.50 per pound per year) = 6,336 + 6,303.50 = 12,640 Therefore, the optimal ordering quantity is 1938 pounds, and the minimum total annual inventory cost is 12,640.

answer:To find the percentage of sales contributed by each product, we first calculate the monthly sales revenue and variable costs for both products. Heritage: - Revenue = Selling price per unit × Units sold = €9.00 × 250 = €2,250 - Variable cost = Variable expenses per unit × Units sold = €5.40 × 250 = €1,350 Gorham: - Revenue = Selling price per unit × Units sold = €10.00 × 200 = €2,000 - Variable cost = Variable expenses per unit × Units sold = €2.00 × 200 = €400 Total monthly sales revenue = Heritage revenue + Gorham revenue = €2,250 + €2,000 = €4,250 Total monthly variable costs = Heritage variable cost + Gorham variable cost = €1,350 + €400 = €1,750 The break-even point in sales units is when total revenue equals total variable costs: Break-even point = Total monthly variable costs / (Selling price per unit - Variable expenses per unit) For Heritage: €1,350 / (€9.00 - €5.40) = 250 units For Gorham: €400 / (€10.00 - €2.00) = 50 units Total break-even units = 250 + 50 = 300 units The weighted average contribution margin is the sum of the contribution margins of each product divided by the total sales revenue: Weighted average contribution margin = (Heritage contribution margin + Gorham contribution margin) / Total sales revenue Heritage contribution margin = (Heritage revenue - Heritage variable cost) / Total sales revenue Gorham contribution margin = (Gorham revenue - Gorham variable cost) / Total sales revenue Heritage contribution margin = (€2,250 - €1,350) / €4,250 = 0.228571429 Gorham contribution margin = (€2,000 - €400) / €4,250 = 0.300943396 Weighted average contribution margin = (0.228571429 + 0.300943396) / 2 = 0.264757413 ≈ 26.48% The percentage of sales for Heritage is: Heritage % of sales = (Heritage units sold / Total break-even units) × Weighted average contribution margin × 100 = (250 / 300) × 26.48% × 100 ≈ 22.07% The percentage of sales for Gorham is: Gorham % of sales = (Gorham units sold / Total break-even units) × Weighted average contribution margin × 100 = (50 / 300) × 26.48% × 100 ≈ 4.41% Thus, the sales percentages are approximately 22.07% for Heritage and 4.41% for Gorham.

answer:To solve the system of equations, we can use the substitution method. Substitute the value of y from the second equation into the first equation: 2x - (-2x + 8) = 4 Simplify the left side of the equation: 2x + 2x - 8 = 4 Combine like terms: 4x - 8 = 4 Add 8 to both sides of the equation: 4x = 12 Divide both sides of the equation by 4: x = 3 Now that we have the value of x, we can substitute it back into either of the original equations to solve for y. Let's use the second equation: y = -2x + 8 y = -2(3) + 8 Simplify the right side of the equation: y = -6 + 8 Combine like terms: y = 2 Therefore, the solution to the system of equations is x = 3 and y = 2. The solution to the system of equations is x = 3 and y = 2.