Homework #6

Important: Read the course outline on the guideline for submitting your solutions

1. 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 which 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 value of p is revenue maximized?

5. The demand function for a product is given by p = -0.03x2- 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 minute. 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 position, 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- 2x2- 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 example.

v

14. Use differentials to approximate 16.1.

15. The weekly profit of a company is given by p(x) = -0.00002x3 + 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.

Important: Read the course outline on the guideline for submitting your solutions

1. 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 which 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 value of p is revenue maximized?

5. The demand function for a product is given by p = -0.03x2- 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 minute. 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 position, 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- 2x2- 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 example.

v

14. Use differentials to approximate 16.1.

15. The weekly profit of a company is given by p(x) = -0.00002x3 + 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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