Related Rates

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

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 long is the diameter when the diameter is changing at a rate of 3 cm/sec?

• .424 cm
• .651 cm
• 1.333 cm
• 1.155 cm

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

A 15 ft ladder is leaning against a vertical wall. The bottom of the ladder is pulled away from the wall at a rate of 2 ft/sec. How fast is the top of the ladder sliding down the wall when the top is 9 ft above the ground?

• 1.2 ft/sec

• 3.6 ft/sec

• 4.8 ft/sec

• 2.4 ft/sec

Two cars start from the same point. Car A travels north at 60 mph and Car B travels east at 80 mph. How fast is the distance between the two cars increasing after 2 hours?

• 100 mph

• 50 mph

• 96 mph

• 72 mph

A spherical balloon is being inflated such that its volume increases at a rate of 5 cm³/sec. How fast is the radius of the balloon increasing when the radius is 10 cm?

• 0.015 cm/sec

• 0.025 cm/sec

• 0.035 cm/sec

• 0.045 cm/sec

A conical tank is being filled with water at a rate of 10 m³/min. The tank has a height of 12 m and a base radius of 3 m. How fast is the water level rising when the water is 4 m deep?

• 0.5 m/min

• 0.3 m/min

• 0.7 m/min

• 0.9 m/min

A man 6 ft tall walks away from a streetlight 18 ft high at a speed of 4 ft/sec. How fast is the tip of his shadow moving when he is 10 ft away from the base of the streetlight?

• 5 ft/sec

• 3 ft/sec

• 6 ft/sec

• 4 ft/sec

A circular plate is being heated such that its radius increases at a rate of 0.2 cm/sec. How fast is the area of the plate increasing when the radius is 5 cm?

• 4π cm²/sec

• 8π cm²/sec

• 6π cm²/sec

• 2π cm²/sec