Stewart 9th Section 2.8: Related Rates

1 Related Rates - Setup · Level 1

If \(V\) is the volume of a cube with edge length \(x\) and the cube expands as time passes, find \(\dfrac{d V}{d t}\) in terms of \(\dfrac{d x}{d t}\).

2 Related Rates - Setup · Level 2

(a) If \(A\) is the area of a circle with radius \(r\) and the circle expands as time passes, find \(\dfrac{d A}{d t}\) in terms of \(\dfrac{d r}{d t}\).

Answer for 2(a)

(b) Suppose oil is spilled and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s, how fast is the area increasing when the radius is 30 m?

Answer for 2(b)

Enter your answer directly below each part above.

3 Related Rates - Basic Geometry · Level 2

Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16 cm²?

4 Related Rates - Basic Geometry · Level 2

The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing when the diameter is 80 mm?

5 Related Rates - Basic Geometry · Level 2

The radius of a spherical ball is increasing at a rate of 2 cm/min. At what rate is the surface area of the ball increasing when the radius is 8 cm?

6 Related Rates - Basic Geometry · Level 2

The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?

7 Related Rates - Basic Geometry · Level 2

A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m³/min. How fast is the height of the water increasing?

8 Related Rates - Basic Geometry · Level 3

The area of a triangle with sides \(a\) and \(b\) and included angle \(\theta\) is \(A = \dfrac{1}{2} a b \sin \theta\). If \(a = 2\) cm, \(b = 3\) cm, and \(\theta\) increases at a rate of \(0.2\) rad/min, how fast is the area increasing when \(\theta = \dfrac{\pi}{3}\)?

9 Related Rates - Implicit · Level 2

If \(x^2 + y^2 = 25\) and \(\dfrac{d x}{d t} = 6\), find \(\dfrac{d y}{d t}\) when \(x = 4\).

10 Related Rates - Implicit · Level 3

If \(x^2 + y^2 + z^2 = 9\), \(\dfrac{d x}{d t} = 5\), and \(\dfrac{d y}{d t} = 4\), find \(\dfrac{d z}{d t}\) when \((x, y, z) = (2, 2, 1)\).

11 Related Rates - Implicit · Level 2

If \(z^2 = x^2 + y^2\), \(\dfrac{d x}{d t} = 2\), and \(\dfrac{d y}{d t} = 3\), find \(\dfrac{d z}{d t}\) when \(x = 5\) and \(y = 12\).

12 Related Rates - Particle Motion · Level 3

A particle moves along the curve \(y = \sqrt{1 + x^3}\). As it reaches the point \((2, 3)\), the \(y\)-coordinate is increasing at a rate of 4 cm/s. How fast is the \(x\)-coordinate of the point changing at that instant?

13 Related Rates - Geometric Change · Level 3

If a snowball melts so that its surface area decreases at a rate of 1 cm²/min, find the rate at which the diameter decreases when the diameter is 10 cm.

14 Related Rates - Geometric Change · Level 2

(a) If \(A\) is the area of a circle with radius \(r\) and the circle expands so that \(\dfrac{d A}{d t} = 3\) cm²/s, find \(\dfrac{d r}{d t}\) when \(r = 1\) cm.

15 Related Rates - Geometric Change · Level 3

How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall?

16 Related Rates - Geometric Change · Level 3

The top of a ladder slides down a vertical wall at a rate of 2 m/s. At the moment when the bottom of the ladder is 4 m from the wall, the bottom slides away from the wall at \(\dfrac{8}{3}\) m/s. How long is the ladder? How fast is the angle between the ladder and the ground changing at that moment when the angle is \(\dfrac{\pi}{4}\) rad?

17 Related Rates - Distance · Level 3

Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?

18 Related Rates - Shadow/Light · Level 3

A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the wall at a speed of 1.6 m/s, how fast is the length of his shadow on the wall decreasing when he is 4 m from the wall?

19 Related Rates - Distance · Level 4

A man starts walking north at 4 ft/s from a point \(P\). Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of \(P\). At what rate are the people moving apart 15 min after the woman starts walking?

20 Related Rates - Applied · Level 3

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s.

(a) At what rate is his distance from second base decreasing when he is halfway to first base?

Answer for 20(a)

(b) At what rate is his distance from third base increasing at the same moment?

Answer for 20(b)

Enter your answer directly below each part above.

21 Related Rates - Applied · Level 3

The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm²/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm²?

22 Related Rates - Applied · Level 3

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?

23 Related Rates - Distance · Level 3

A woman standing on a cliff drops a stone. The distance the stone has fallen after \(t\) seconds is \(d = 4.9 t^2\) meters. One second after the first stone is dropped, she drops a second stone. One second after that, how fast is the distance between the two stones changing?

24 Related Rates - Distance · Level 4

Two men stand 10 m apart on level ground, each at the edge of a cliff. One man drops a stone; one second later the other drops a stone. One second after the second stone is dropped, how fast is the distance between the two stones changing?

25 Related Rates - Applied · Level 4

Water is leaking out of an inverted conical tank at a rate of 10,000 cm³/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.

26 Related Rates - Particle Motion · Level 4

A particle moves along the curve \(y = 2 \sin\left(\dfrac{\pi x}{2}\right)\). As the particle passes through the point \(\left(\dfrac{1}{3}, 1\right)\), its \(x\)-coordinate increases at a rate of \(\sqrt{10}\) cm/s. How fast is the distance from the particle to the origin changing at that instant?

27 Related Rates - Applied · Level 4

A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at a rate of 0.2 m³/min, how fast is the water level rising when the water is 30 cm deep?

28 Related Rates - Applied · Level 3

A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft³/min, how fast is the water level rising when the water is 6 inches deep?

29 Related Rates - Applied · Level 3

Gravel is being dumped from a conveyor belt at a rate of 30 ft³/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

30 Related Rates - Applied · Level 4

A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A cross-section is shown in the figure. If the pool is being filled with water at a rate of 0.8 ft³/min, how fast is the water level rising when the depth at the deepest point is 5 ft?

31 Related Rates - Geometric Change · Level 2

The sides of an equilateral triangle are increasing at a rate of 10 cm/min. At what rate is the area of the triangle increasing when the sides are 30 cm long?

32 Related Rates - Applied · Level 3

A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?

33 Related Rates - Distance · Level 4

A car travels north on a straight road at 20 m/s and a drone flies east at 6 m/s at an elevation of 25 m, both passing through the same point on the ground at different times. When the drone is directly above the car's position, how fast is the distance between the car and the drone changing 5 s later?

34 Related Rates - Geometric Change · Level 2

The minute hand of a clock has length \(r\) cm. Find the rate at which it sweeps out area as a function of \(r\).

35 Related Rates - Applied · Level 3

How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall?

36 Related Rates - Applied · Level 3

According to the model in Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for describing the speed of the top of the ladder for small values of \(y\)?

37 Related Rates - Applied · Level 3

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure \(P\) and volume \(V\) satisfy the equation \(P V = C\), where \(C\) is a constant. Suppose that at a certain instant the volume is 600 cm³, the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPa/min. At what rate is the volume decreasing at this instant?

38 Related Rates - Applied · Level 4

A hemispherical basin has a diameter of 60 cm and is being filled with water at a rate of 2 L/min. Find the rate at which the water is rising when the water is half-full. [Use the fact that the volume of water in the basin is \(V = \pi\left(r h^2 - \dfrac{1}{3} h^3\right)\) when the water depth is \(h\) and the radius of the hemisphere is \(r\).]

39 Related Rates - Applied · Level 3

When two resistors with resistances \(R_1\) and \(R_2\) are connected in parallel, their combined resistance \(R\) is given by \(\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2}\). If \(R_1\) is increasing at a rate of 0.3 \(\Omega\)/s and \(R_2\) is increasing at a rate of 0.2 \(\Omega\)/s, how fast is \(R\) changing when \(R_1 = 80\) \(\Omega\) and \(R_2 = 100\) \(\Omega\)?

40 Related Rates - Applied · Level 3

When gas expands in a cylinder with radius \(r\), the pressure at any given time is a function of the volume: \(P = P(V)\). The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: \(F = \pi r^2 P\). Show that the work done by the gas when the volume expands from volume \(V_1\) to volume \(V_2\) is \(W = \displaystyle\int_{V_1}^{V_2} P d V\). [The adiabatic expansion satisfies \(P V^{1.4} = C\). At a certain instant the volume is 400 cm³ and the pressure is 80 kPa and is decreasing at a rate of 10 kPa/min. At what rate is the volume increasing at that instant?]

41 Related Rates - Applied · Level 4

Two straight roads diverge at an angle of 60°. Two cars leave the intersection at the same time, the first traveling down one road at 40 mi/h and the second traveling down the other road at 60 mi/h. How fast is the distance between the cars changing after half an hour? [Hint: Law of Cosines.]

42 Related Rates - Applied · Level 5

Brain weight \(B\) as a function of body weight \(W\) in fish has been modeled by the power function \(B = 0.007 W^{\dfrac{2}{3}}\). A model for body weight as a function of body length \(L\) is \(W = 0.12 L^{2.53}\). If, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was the brain weight increasing when the average length was 18 cm?

43 Related Rates - Applied · Level 4

Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2°/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60°? [Hint: Law of Cosines.]

44 Related Rates - Applied · Level 4

Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley \(P\). The point \(Q\) is on the floor, 12 ft directly beneath \(P\), and between the carts. Cart A is being pulled away from \(Q\) at a speed of 2 ft/s. How fast is cart B moving toward \(Q\) at the moment when cart A is 5 ft from \(Q\)?

45 Related Rates - Applied · Level 3

A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the determine the mechanism through which the camera angle changes. Find the rates of change of the angle of elevation and the camera-to-rocket distance at the instant the rocket is 3000 ft above the launch pad and the rocket is rising at the rate of 600 ft/s.

(a) How fast is the distance from the camera to the rocket changing at that instant?

Answer for 45(a)

(b) If the camera is always directed at the rocket, how fast is the camera's angle of elevation changing at that same instant?

Answer for 45(b)

Enter your answer directly below each part above.

46 Related Rates - Shadow/Light · Level 3

A lighthouse is located on a small island 3 km away from the nearest point \(P\) on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from \(P\)?

47 Related Rates - Applied · Level 4

A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is \(\dfrac{\pi}{3}\), this angle is decreasing at a rate of \(\dfrac{\pi}{6}\) rad/min. How fast is the plane traveling?

48 Related Rates - Applied · Level 3

A Ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level?

49 Related Rates - Applied · Level 4

A plane flying with a constant speed of 300 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30°. At what rate is the distance from the plane to the radar station increasing one minute later?

50 Related Rates - Distance · Level 4

Two people start from the same point. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How fast is the distance between the people changing after 15 minutes?

51 Related Rates - Distance · Level 5

A runner sprints around a circular track of radius 100 m at a constant speed of 7 m/s. The runner's friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m?

52 Related Rates - Applied · Level 5

The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o'clock?

53 Related Rates - Proof · Level 4

Suppose that the volume \(V\) of a rolling snowball increases so that \(\dfrac{d V}{d t}\) is proportional to the surface area of the snowball at time \(t\). Show that the radius \(r\) increases at a constant rate, that is, \(\dfrac{d r}{d t}\) is constant.

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