1. Let z = x2 + y2, where x and y are increasing at the constant rates of 2 units per second and 3 units per second, respectively. What is the set of points for which the rate of change of z is zero?

 

A. {(x,y): 3y = -2x}

B. {(x,y): 3y = 2x}

C. {(x,y): x = y = 0}

D. {(x,y): 2y = -3x}

E. {(x,y): 2y = 3x}

 

solution

 

 

2. A water tank in the shape of a right circular cone has a height of 10 feet. The top rim of the tank is a circle with a radius of 4 feet. If water is being pumped into the tank at the rate of 2 cubic feet per minute, what is the rate of change of the water depth, in feet per minute, when the depth is 5 feet?

(The volume V of a right circular cone is V = 1/3 p r2h, where r is its radius and h is its height.)

 

A. 1/2p

B. 1/p

C. 3/2p

D. 2/p

E. 5/2p

 

solution

 

3. A cube of ice melts without changing shape at the uniform rate of 4 cm3/min. What is the rate of change of the surface area of the cube, in cm2/min, when the volume of the cube is 125 cm3?

 

A. -4

B. 16/5

C. 16/6

D. 60/19

E. 16/5

 

solution

 

4. Let x be the length of one of the equal sides of an isosceles triangle, and let q be the angle between them. If x is increasing at the rate 1/12 m/hr, and q is increasing at the rate of p /180 radians/hr, then at what rate, in m2/hr, is the area of the triangle increasing when x = 12 m and q = p /4?

 

A. 36(21/2)

B. (73/2)(21/2)

C. (31/2/2)+(p /5)

D. 21/2((1/2)+(1/5)p )

E. 21/2(6+(1/5)p )

 

solution

 

5. The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. What is the rate of change of the volume of the cylinder, in cm3/min, when the radius is 2 cm and the height is 3 cm? (Note: The volume of a right circular cylinder is V = p r2h.)

 

A. -2p

B. -8p /5

C. -3p /5

D. 2p /5

E. 2p

 

solution

 

6. The height h of a right circular cone is 20 cm and is decreasing at the rate of 4 cm/sec. At the same time, the radius r is 10 cm and is increasing at the rate of 2 cm/sec. What is the rate of change of the volume in cm3/sec? (Note: The volume of a right circular cone is V = 1/3p r2h.)

 

A. -400p

B. -400p /3

C. 0

D. 400p/3

E. 400p

 

solution