63496 – Homework #6Important: Read the course outline on the

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Homework #6Important: Read the course outline on the guideline for submitting your solutions1. The number of sales per week of an $80 pair of jeans is 25. The number of weekly sales goes up to 45 when the price is reduced by 30%.(a) Find the percentage change in demand.(b) Find the average elasticity of demand.2. The price-demand equation of a product is given by x = 1000 – 2p2.(a) Find the elasticity of demand at p = 10 and p = 15. Interpret your answers.(b) Find the price that maximizes the revenue.3. Consider the price-demand equation p = 2000 – 0.04x, 0 = x = 10000.(a) Find the elasticity of demand E(p).(b) Compute E(500) and E(1200). Interpret the result.(c) If the price per unit is $1200 and is lowered by 10%, by what percentage will the demand change?(d) Find the price that maximizes the revenue.4. The demand for a particular commodity when sold at a price of p dollars is given by the function D(p) = 4000e-0.02p.(a) Find the price elasticity of demand function and determine the values of p for the demand is elastic, inelastic, and of unitary elasticity.(b) If the price is increased by 3% from $12, what is the approximate effect on demand?(c) Find the revenue R(p) obtained by selling q units at p dollars per unit. For what of p is revenue maximized?5. The demand function for a product is given by p = -0.03×2- 0.1x + 21, where p is the unit price and x is the quantity demanded. Find the elasticity of demand E when x = 10.6. The height of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/min. How quickly is the base of the triangle changing when the height is 10 cm and the area is 100 cm2?7. A manufacturer has found that the cost C and revenue R in one month are related by the equation 267C = R2 + 4700. Find the rate of change of revenue with respect to time when the cost is changing by 12 units per month and the monthly cost is 32 units.8. A 26 m ladder is placed against a building. The base of the ladder is slipping away from the building at a rate of 1.5 m per . Find the rate at which the top of the ladder is sliding down the building at the instant when the bottom of the ladder is 10 m from the base of the building.9. A balloon is taking off from the ground on on a vertical line. A viewer, standing 100 feet away from the balloon’s take-off , observes that the elevation angle is increasing at the rate of radians per minute. What is the speed of the balloon when the angle of elevation is ?10. Suppose that water is being emptied from a spherical tank of radius 10 ft. If the depth of water in the tank is 5 ft and is decreasing at the rate of 3 ft/sec, at what rate is the radius r of the top surface of the water decreasing?11. Consider the function f(x) = x3- 2×2- 3x + 10.(a) Find the equation y = L(x) of the line tangent to y = f(x) at the point (-1,10).(b) Find f(-1),L(-1),f(-0.98), and L(-0.98).12. Given the function y = 1 – x3.(a) Find ?x and ?y when x changes from 1 to 1.02. (b) Find ?x and ?y when x changes from 1 to 0.98.13. Given the function y = 1 – x3.(a) Find the differential dy when x changes from 1 to 1.02.(b) Compare your result with ? found in the previous .v14. Use differentials to approximate 16.1.15. The weekly profit of a company is given by p(x) = -0.00002×3 + 40x – 90 (in thousands of dollars), where x is the number of produced and sold items (in thousands). Use differentials to estimate the change in profit when quantity demanded changes from 5000 to 5500 items.

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