Exercise set 4.2
- 1. EXERCISE SET 4.2 1. A store can sell 100 bags of ice at a price of $2 per bag and 80 bags at $2.50. Assume the price-demand function is linear. (a) Find the price-demand function. (b) Find the revenue function. (c) Find the revenue and rates of change of revenue if x = $2 and x = $3. Explain the results. (d) Find the price that would produce the maximum revenue and the maximum revenue. 2. A hardware store can sell 60 saws at a price of $8 per saw and 70 saws at $6. Assume the price-demand function is linear. (a) Find the price-demand function. (b) Find the revenue function. (c) Find the revenue and rates of change of revenue if x = $7 and x = $14. Explain the results. (d) Find the price that would produce the maximum revenue and the maximum revenue. 3. (a) Express the volume V of a sphere as a function of its surface area S. (b) Find dV/dS and evaluate it at S = 16π and S = 144π. (c) What can you conclude? 4. (a) Express the area A of a circle as a function of its circumference C. (b) Find dA/dC and evaluate it at C = 2π and C = 10π. (c) What can you conclude? 5. (a) Express the volume V of a rectangular parallelepiped, whose dimensions are x by x by 2x, as a function of its surface area S. (b) Find dV/dS and evaluate it at S = 10 and S = 90. (c) What can you conclude? 6. (a) Express the volume V of a rectangular parallelepiped, whose dimensions are x by 2x by 3x, as a function of its surface area S. (b) Find dV/dS and evaluate it at S = 22 and S = 88. (c) What can you conclude?
- 2. 7. A ladder 20 feet long leans on a house touching it h feet above the ground. Let be the angle the ladder makes with the ground. (a) Express h in terms of . Find the rate of change of h with respect to . What are the units of this rate of change? What is the rate of change when = 60 ? (b) Approximate the height gained when the angle increases from 60 to 62 . 8. A ladder 8 meters long leans on a house touching it h meters above the ground. Let be the angle the ladder makes with the house. (a) Express h in terms of . Find the rate of change of h with respect to . What are the units of this rate of change? What is the rate of change when = 60 ? (b) Approximate the height lost when the angle increases from 30 to 33 . 9. The electric force between two charged particles separated by x meters is given by 24 F= where F is in Newtons. If the particles are 5 meters apart, find the approximate change x2 in the force if they are moved 10 centimeters closer. 10. The force of gravity between two masses separated by x kilometers is given by 36 F= where F is in pounds. If the masses are 2 km apart, find the approximate change in the x2 force if they are moved 20 meters closer.