A construction company is calculating the total cost of a project. The cost of materials is given by M(t) = 2t2 + 500, and the cost of labor is given by L(t) = 3t3 - 100t, where t is the time in weeks. What is the rate of change of the total cost with respect to time?

• 6t2 + 4t + 400

• 5t5 - 100t + 500

• 9t2 + 4t - 100

• 6t2 + 4t

The velocity of a car is given by v(t) = 5t2. If the car's mass is 1500 kg, what is the rate of change of the car's momentum (p = mv) with respect to time?

• 15000t

• 10t

• 7500t2

• 5t

A farmer is calculating the area of a rectangular field. The length is increasing at a rate of 2 m/s, and the width is increasing at a rate of 3 m/s. If the length is given by l(t) and the width by w(t), what rule helps find the rate of change of the area?

• Sum Rule

• Quotient Rule

• Product Rule

• Chain Rule

The height of a projectile is given by h(t) = -5t2 + 20t + 10. What is the instantaneous velocity of the projectile at t = 2 seconds?

• -20 m/s

• 0 m/s

• 10 m/s

• 20 m/s

A company's revenue is given by R(x) = (x + 5)(2x - 1), where x is the number of units sold. What is the rate of change of revenue with respect to the number of units sold?

• 4x+9

• 2x-1

• x+5

• 2

The cost of producing x items is C(x) = 5x3 + 4x2 - 2x + 10. What is the marginal cost (the rate of change of cost) when x = 5?

• 340

• 10

• 4

• 325

A population of bacteria grows according to the function P(t) = 100e(0.2t), where t is time in hours. What is the rate of growth of the bacteria population?

• 100e(0.2t)

• 0.2t

• 20e(0.2t)

• 0.2

The temperature T of a metal rod varies along its length x according to the function T(x) = 2x^2 - 5x + 100. What is the rate of change of temperature with respect to length at x = 3?

• 4x - 5

• 100

• 7

• 2x2

A spherical balloon is being inflated. The volume of a sphere is given by V = (4/3)πr3. If the radius r is increasing at a rate of 2 cm/s, what rule helps find the rate of change of the volume?

• Product Rule

• Quotient Rule

• Chain Rule

• Sum Rule

The price of a stock is modeled by P(t) = (t2 + 1) / (t + 3), where t is the time in days. What rule is used to find the rate of change of the stock price?

• Product Rule

• Quotient Rule

• Chain Rule

• Sum Rule