Calculus-Related Rates Problems

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

• 3 ft/sec
• -7.2 ft/sec
• 7.2 ft/sec
• 12

Tags

CCSS.8.G.B.8

Chris is sitting on the edge of a dock tossing rocks into the water. As each rock hits the water, small circles appear traveling outward from the point of impact. The radius of the circle is changing at a rate of 5 in/sec.  How fast is the area of the outer circle changing when the diameter is 8 in?

• 80pi in/sec
• 20pi in/sec
• 60pi in/sec
• 4opi in/sec

Tags

CCSS.HSA.SSE.A.1

CCSS.HSG.GMD.A.1

Brandon is starting to clean up after a birthday party. He begins deflating each spherical balloon by puncturing a hole in each. The air leaves the balloon at a constant rate of 2 cm³/sec.  How fast is the diameter changing when the diameter is 8 cm?

• -1/(16pi) cm/sec
• -1/16 cm/sec
• -1/(4pi) cm/sec
• 1/(4pi) cm/sec

Tags

CCSS.HSG.GMD.A.3

Louisa and Karis were each dropped off at the same bus stop. Louisa’s bus drops her off at 3:30 whereas Karis is dropped off ten minutes later. Louisa runs home at a constant rate of 6 mph and Karis runs home at 3 mph. Louisa lives north of the bus stop and Karis lives to the east.  How fast is the distance between them changing at 4:00?

• 6.512 mph
• 7.115 mph
• 6.708 mph
• 6.641 mph

Tags

CCSS.HSA.CED.A.2

CCSS.HSA.SSE.A.1

CCSS.HSA.REI.C.6

CCSS.HSF.BF.A.1

A baseball diamond is a square with a side of 90 ft. Tucker hits the ball and runs towards first base at 16 ft/sec.  How fast is the distance between second base and Tucker changing when he is 30 ft from first base?

• 8.875 ft/sec
• 5.060 ft/sec
• -5.060 ft/sec
• -8.875 ft/sec

Tags

CCSS.HSA.CED.A.2

(3 min) The area of a circular region is increasing at a rate of 96 π square meters per second. When the area of the region is 64 π square meters, how fast, in meters per second, is the radius of the region increasing?

• 6
• 8
• 16
• 4√3

Tags

CCSS.7.G.B.4

(3 min) The radius r of a sphere is increasing at the uniform rate of 0.3 inches per second. At the instant when the surface area S becomes 100 π square inches, what is the rate of increase, in cubic inches per second, in the volume V? SA = 4 π r² and V = (4/3) π r³

• 30π
• 10π
• 12π
• 25π

Tags

CCSS.HSG.GMD.A.3

A certain medical procedure requires that a balloon be inserted into the stomach and then inflated. Model the shape of the balloon by a sphere of radius r. If r is increasing at the rate of 0.3 cm/min, how fast is the volume changing when the radius is 4 cm?

• 15.08 cm³/min
• 268.08 cm³/min
• 60.32 cm³/min
• 6.03 cm³/min

The radius r of a sphere is increasing at a rate of 2 inches per minute.  Find the rate of change of the volume when r=6 inches.

• C=2πr
• A=πr²
• V=(4/3)πr³
• S=4πr²

Tags

CCSS.HSG.GMD.A.3

All edges of a cube are expanding at a rate of 3 centimeters per second.  How fast is the volume changing each edge is 1 centimeter.

• A=l*w
• V=3s²
• A=s²
• V=s³

Tags

CCSS.HSG.GMD.A.3

CCSS.HSG.GMD.A.1