UHS AP Calculus AB | Related Rates Related to You

1. Taylor and Holly are blowing up balloons for Hoedown. The diameter of a balloon increases at a rate of 10 cm/sec. At what rate must they blow air into the balloon when the diameter measures 4 cm? Assume that the balloons are spherical ). cm3/sec

2. Asa and Seth leave Tuttle Stadium from the same point at the same time. If Asa runs south at 4 meters per second and Seth runs west at 3 meters per second, how fast will the distance between Asa and Seth be changing after 10 seconds? 5 m/sec

3. Suppose Jay is pumping water into a tank (in the shape of an inverted right circular cone, obviously) at a rate of 1600 ft3/min. Find the rate at which the radius is changing when the height of the water is 6.5 ft, the radius is 3 feet, and the height is changing at a rate of 100 feet per minute . 16.099 ft/min

4. SanchezCorp Industries has recently gotten into the business of making off-brand Snuggies®. They hire Professor Molly Thompson to calculate the revenue and cost of their best-selling product, the Snuggle-Wuggle®. Professor Thompson finds that the revenue is and the cost is , where x is the number of Sunggle-Wuggles® produced each week. If SanchezCorp produces 2500 Snuggle-Wuggles® this week, find the marginal profit. Marginal profit is the approximate profit of producing one more unit and is given by the derivative of profit. Keep in mind: profit P = R − C. $5/Snuggle-Wuggle®

5. The talented Emily Beach is flying a kite at a constant height of 400 meters. The kite is moving horizontally at a rate of 30 m/sec. How fast must she unwind the string when the kite is 500 m away from her? 18 m/sec

6. A ladder 15 feet tall leans against a vertical wall of a home. Angie tries to put some Christmas lights up using this ladder. Unfortunately, just as she finishes the final touches of her masterpiece, the ladder starts to fall from under her (Glory denies all involvement). If the bottom of the ladder is pulled away horizontally from the house at 4 ft/sec, how fast is the top of the ladder/Angie sliding down the wall when the bottom of the ladder is 9 feet from the wall? -3 ft/sec

7. A streetlight is 15 feet above the sidewalk. Parker (who we’ll say is 7 feet tall) walks away from the light at a rate of 5 feet per second. Determine the rate at which Parker’s shadow is changing.

35/8 ft/sec

8. Hayley and Bethany leave a parking lot at the same time; Hayley travels north and Bethany travels west. At time t, Hayley’s velocity can be modeled by , and at time t=2 minutes, she is 400 meters north of the parking lot. At the same time (t=2), Bethany is traveling at a rate of 100 meters per minute and she is 300 meters west of the parking lot. Find the rate, in meters per minute, at which the distance between Hayley and Bethany is changing at t=2. 160 m/min

UHS AP Calculus AB | Related Rates Related to You

1. Chris and Jesse are blowing up balloons for Hoedown. The diameter of a balloon increases at a rate of 10 cm/sec. At what rate must they blow air into the balloon when the diameter measures 4 cm? Assume that the balloons are spherical ). cm3/sec

2. Tyler and Ayo leave Tuttle Stadium from the same point at the same time. If Tyler runs south at 4 meters per second and Ayo runs west at 3 meters per second, how fast will the distance between Tyler and Ayo be changing after 10 seconds? 5 m/sec

3. Suppose Abi is pumping water into a tank (in the shape of an inverted right circular cone, obviously) at a rate of 1600 ft3/min. Find the rate at which the radius is changing when the height of the water is 6.5 ft, the radius is 3 feet, and the height is changing at a rate of 100 feet per minute . 16.099 ft/min

4. BroadhurstCorp Industries has recently gotten into the business of making off-brand Snuggies®. They hire Professor Madalyn Grass to calculate the revenue and cost of their best-selling product, the Snuggle-Wuggle®. Professor Grass finds that the revenue is and the cost is , where x is the number of Sunggle-Wuggles® produced each week. If BroadhurstCorp produces 2500 Snuggle-Wuggles® this week, find the marginal profit. Marginal profit is the approximate profit of producing one more unit and is given by the derivative of profit. Keep in mind: profit P = R − C. $5/Snuggle-Wuggle®

5. The talented Aysha Trivedi is flying a kite at a constant height of 400 meters. The kite is moving horizontally at a rate of 30 m/sec. How fast must she unwind the string when the kite is 500 m away from her? 18 m/sec

6. A ladder 15 feet tall leans against a vertical wall of a home. Miles tries to put some Christmas lights up using this ladder. Unfortunately, just as he finishes the final touches of his masterpiece, the ladder starts to fall from under him (Emily denies all involvement). If the bottom of the ladder is pulled away horizontally from the house at 4 ft/sec, how fast is the top of the ladder/Miles sliding down the wall when the bottom of the ladder is 9 feet from the wall? -3 ft/sec

7. A streetlight is 15 feet above the sidewalk. Alyssa, who is clearly 7 feet tall, walks away from the light at a rate of 5 feet per second. Determine the rate at which Alyssa’s shadow is changing.

35/8 ft/sec

8. Dawniece and Nhi leave a parking lot at the same time; Dawniece travels north and Nhi travels west. At time t, Dawniece’s velocity can be modeled by , and at time t=2 minutes, she is 400 meters north of the parking lot. At the same time (t=2), Nhi is traveling at a rate of 100 meters per minute and she is 300 meters west of the parking lot. Find the rate, in meters per minute, at which the distance between Dawniece and Nhi is changing at t=2. 160 m/min