Related Rates Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) Suppose that the radius and surface area of a sphere are differentiable functions of Write an equation that relates to

(2) If , and are lengths of the edges of a rectangular box, the common length of the box's diagonals is (a) Assuming that , and are differentiable functions of how is related to and

(3) The length of a rectangle is decreasing at the rate of while the width is increasing at the rate of When and find the rates of change of (a) the area, (b) the perimeter, (c) the lengths of the diagonals of the rectangle. Which of these quantities are decreasing and which are increasing?

(4) A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of (a) How fast is the top of the ladder sliding down the wall then? (b) At what rate is the area of the triangle formed by the ladder, wall, and ground changing then? (c) At what rate is the angle between the ladder and the ground changing then?

(5) Two commercial airplanes are flying at 40,000 ft along straight-line courses that intersect at right angles. Plane is approaching the intersection point at a speed of 442 knots. Plane is approaching the intersection at 481 knots. At what rate is the distance between the planes changing when is nautical miles from the intersection point and is 12 nautical miles from the intersection point?

(6) Sand falls from a conveyor belt at the rate of onto the top of a conical pile. The height of the pile is always three-eights of the base diameter. How fast are the (a) height and (b) radius changing when the pile is high?

(7) Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to is surface area. Show that under thee circumstances the drop's radius increase at a constant rate.

(8) A balloon is rising vertically above a level straight road at a constant rate of Just when the balloon is above the ground, a bicycle moving at a constant rate of passes under it. How fast is the distance between the bicycle and balloon increasing 3 sec later?

(9) The coordinates of a particle in the metric -plane are differentiable functions of time with and How fast is the particle's distance from the origin changing as it passes through the point

(10) A man 6 ft all walks at a rate of toward a streetlight that is 16 ft above the ground. At what rate s the tip of his shadow moving? At what rate is the length of his shadow changing when he is 10 ft from the base of the light?

(11) All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each side is (a) 1 centimeter and (b) 10 centimeters?

(12) A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep?

(13) A balloon rises at a rate of 3 meters per second from a point on the ground 30 meters from an observer. What is the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground?

(14) A trough is 10 feet long and its ends have the shape of an isosceles triangles that are 3 feet across at the top and have a height of 1 foot. If the trough is filled with water at a rate of how fast does the water level rise when the water is 6 inches deep?

(15) A water trough is 10 m long and a cross section has the shape of an isosceles trapezoid that is 30 centimeters wide at the bottom, 80 centimeters wide at the top, and has height 50 centimeters. If it is being filled with water at the rate of how fast is the water level rising when the water is 30 centimeters deep?

Cite this as:
Related Rates Homework
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/related-rates-homework.html