Читать книгу Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice) - Patrick Jones - Страница 22

432– 445 Solve the related-rates problem. Give an exact answer unless otherwise stated.

432. If V is the volume of a sphere of radius r and the sphere expands as time passes, find in terms of .

433. A pebble is thrown into a pond, and the ripples spread in a circular pattern. If the radius of the circle increases at a constant rate of 1 meter per second, how fast is the area of the circle increasing when the radius is 4 meters?

434. If and , find when .

435. If , , and , find when and .

436. Two sides of a triangle are 6 meters and 8 meters in length, and the angle between them is increasing at a rate of 0.12 radians per second. Find the rate at which the area of the triangle is increasing when the angle between the sides is . Round your answer to the nearest hundredth.

437. A ladder 8 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 3 feet per second, how fast is the angle between the top of the ladder and the wall changing when the angle is radians?

438. The base of a triangle is increasing at a rate of 2 centimeters per minute, and the height is increasing at a rate of 4 centimeters per minute. At what rate is the area changing when centimeters and centimeters?

439. At noon, Ship A is 150 kilometers east of Ship B. Ship A is sailing west at 20 kilometers per hour, and Ship B is sailing north at 35 kilometers per hour. How quickly is the distance between them changing at 3 p.m.? Round your answer to the nearest hundredth.

440. A particle moves along the curve . As the particle passes through the point , the x coordinate is increasing at a rate of 5 centimeters per second. How quickly is the distance from the particle to the origin changing at this point? Round your answer to the nearest hundredth.

441. Two people start walking from the same point. One person walks west at 2 miles per hour, and the other walks southwest (at an angle 45° south of west) at 4 miles per hour. How quickly is the distance between them changing after 40 minutes? Round your answer to the nearest hundredth.

442. A trough is 20 feet long, and its ends are isosceles triangles that are 5 feet across the top and have a height of 2 feet. If the trough is being filled with water at a rate of 8 cubic feet per minute, how quickly is the water level rising when the water is 1 foot deep?

443. An experimental jet is flying with a constant speed of 700 kilometers per hour. It passes over a radar station at an altitude of 2 kilometers and climbs at an angle of 45°. At what rate is the distance from the plane to the radar station increasing 2 minutes later? Round your answer to the hundredths place.

444. A lighthouse is located on an island 5 kilometers away from the nearest point P on a straight shoreline, and the light makes 6 revolutions per minute. How fast is the beam of light moving along the shore when it’s 2 kilometers from P?

445. Gravel is being dumped into a pile that forms the shape of a cone whose base diameter is twice the height. If the gravel is being dumped at a rate of 20 cubic feet per minute, how fast is the height of the pile increasing when the pile is 12 feet high?