
# 4.1E: Exercises for Section 4.1


In exercises 1 - 3, find the quantities for the given equation.

1) Find $$\frac{dy}{dt}$$ at $$x=1$$ and $$y=x^2+3$$ if $$\frac{dx}{dt}=4.$$

$$\dfrac{dy}{dt} = 8$$

2) Find $$\frac{dx}{dt}$$ at $$x=−2$$ and $$y=2x^2+1$$ if $$\frac{dy}{dt}=−1.$$

3) Find $$\frac{dz}{dt}$$ at $$(x,y)=(1,3)$$ and $$z^2=x^2+y^2$$ if $$\frac{dx}{dt}=4$$ and $$\frac{dy}{dt}=3$$.

$$\dfrac{dz}{dt} = \frac{13}{\sqrt{10}}$$

In exercises 4 - 15, sketch the situation if necessary and used related rates to solve for the quantities.

4) [T] If two electrical resistors are connected in parallel, the total resistance (measured in ohms, denoted by the Greek capital letter omega, $$Ω$$) is given by the equation $$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}.$$ If $$R_1$$ is increasing at a rate of $$0.5Ω/\text{min}$$ and $$R_2$$ decreases at a rate of $$1.1Ω/\text{min}$$, at what rate does the total resistance change when $$R_1=20Ω$$ and $$R_2=50Ω/\text{min}$$?

5) A $$10$$-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of $$2$$ ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is $$5$$ ft from the wall?

$$2\sqrt{3}$$ ft/sec

6) A $$25$$-ft ladder is leaning against a wall. If we push the ladder toward the wall at a rate of $$1$$ ft/sec, and the bottom of the ladder is initially $$20$$ ft away from the wall, how fast does the ladder move up the wall $$5$$ sec after we start pushing?

7) Two airplanes are flying in the air at the same height: airplane A is flying east at $$250$$ mi/h and airplane B is flying north at $$300$$ mi/h. If they are both heading to the same airport, located $$30$$ miles east of airplane A and $$40$$ miles north of airplane B, at what rate is the distance between the airplanes changing?

The distance is decreasing at $$390$$ mi/h.

8) You and a friend are riding your bikes to a restaurant that you think is east; your friend thinks the restaurant is north. You both leave from the same point, with you riding at $$16$$ mph east and your friend riding $$12$$ mph north. After you traveled $$4$$ mi, at what rate is the distance between you changing?

9) Two buses are driving along parallel freeways that are $$5$$ mi apart, one heading east and the other heading west. Assuming that each bus drives a constant $$55$$ mph, find the rate at which the distance between the buses is changing when they are $$13$$ mi apart (as the crow flies), heading toward each other.

The distance between them shrinks at a rate of $$\frac{1320}{13}≈101.5$$ mph.

10) A $$6$$-ft-tall person walks away from a $$10$$-ft lamppost at a constant rate of $$3$$ ft/sec. What is the rate that the tip of the shadow moves away from the pole when the person is $$10$$ ft away from the pole?

11) Using the previous problem, what is the rate at which the tip of the shadow moves away from the person when the person is $$10$$ ft from the pole?

$$\frac{9}{2}$$ ft/sec

12) A $$5$$-ft-tall person walks toward a wall at a rate of $$2$$ ft/sec. A spotlight is located on the ground $$40$$ ft from the wall. How fast does the height of the person’s shadow on the wall change when the person is $$10$$ ft from the wall?

13) Using the previous problem, what is the rate at which the shadow changes when the person is $$10$$ ft from the wall, if the person is walking away from the wall at a rate of $$2$$ ft/sec?

It grows at a rate $$\frac{4}{9}$$ ft/sec

14) A helicopter starting on the ground is rising directly into the air at a rate of $$25$$ ft/sec. You are running on the ground starting directly under the helicopter at a rate of $$10$$ ft/sec. Find the rate of change of the distance between the helicopter and yourself after $$5$$ sec.

15) Using the previous problem, assuming the helicopter is again rising at a rate of $$25$$ ft/sec and you are running on the ground starting directly under the helicopter at a rate of $$10$$ ft/sec, what is the rate at which the distance between you and the helicopter is changing when the helicopter has risen to a height of $$60$$ ft in the air, assuming that, initially, it was $$30$$ ft above you?

The distance is increasing at $$\frac{135\sqrt{26}}{26}$$ ft/sec

In exercises 16 - 24, draw and label diagrams to help solve the related-rates problems.

16) The side of a cube increases at a rate of $$\frac{1}{2}$$ m/sec. Find the rate at which the volume of the cube increases when the side of the cube is $$4$$ m.

17) The volume of a cube decreases at a rate of $$10 \text{ m}^3$$/sec. Find the rate at which the side of the cube changes when the side of the cube is $$2$$ m.

$$−\frac{5}{6}$$ m/sec

18) The radius of a circle increases at a rate of $$2$$ m/sec. Find the rate at which the area of the circle increases when the radius is $$5$$ m.

19) The radius of a sphere decreases at a rate of $$3$$ m/sec. Find the rate at which the surface area decreases when the radius is $$10$$ m.

$$240π \,\text{m}^2\text{/sec}$$

20) The radius of a sphere increases at a rate of $$1$$ m/sec. Find the rate at which the volume increases when the radius is $$20$$ m.

21) The radius of a sphere is increasing at a rate of $$9$$ cm/sec. Find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.

$$\frac{1}{2\sqrt{π}}$$ cm

22) The base of a triangle is shrinking at a rate of $$1$$ cm/min and the height of the triangle is increasing at a rate of $$5$$ cm/min. Find the rate at which the area of the triangle changes when the height is $$22$$ cm and the base is $$10$$ cm.

23) A triangle has two constant sides of length $$3$$ ft and $$5$$ ft. The angle between these two sides is increasing at a rate of $$0.1$$ rad/sec. Find the rate at which the area of the triangle is changing when the angle between the two sides is $$π/6.$$

The area is increasing at a rate $$\frac{3\sqrt{3}}{8}\,\text{ft}^2\text{/sec}$$.

24) A triangle has a height that is increasing at a rate of $$2$$ cm/sec and its area is increasing at a rate of $$4 \,\text{cm}^2\text{/sec}$$. Find the rate at which the base of the triangle is changing when the height of the triangle is $$4$$ cm and the area is $$20 \,\text{cm}^2$$.

In exercises 25 - 27, consider an inverted right cone that is leaking water.  (Inverted means the cone's point is facing down, like a funnel.) The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft.

25) How fast does the depth of the water change when the water is $$10$$ ft high if the cone leaks water at a rate of $$10 \,\text{ft}^3\text{/min}$$?

The depth of the water decreases at $$\frac{128}{125π}$$ ft/min.

26) Find the rate at which the surface area of the water changes when the water is $$10$$ ft high if the cone leaks water at a rate of $$10 \,\text{ft}^3\text{/min}$$.

27) If the water level is decreasing at a rate of $$3$$ in/min when the depth of the water is $$8$$ ft, determine the rate at which water is leaking out of the cone.

The volume is decreasing at a rate of $$\frac{25π}{16}\,\text{ft}^3\text{/min}$$.

28) A vertical cylinder is leaking water at a rate of $$1 \,\text{ft}^3\text{/sec}$$. If the cylinder has a height of $$10$$ ft and a radius of $$1$$ ft, at what rate is the height of the water changing when the height is $$6$$ ft?

29) A cylinder is leaking water but you are unable to determine at what rate. The cylinder has a height of $$2$$ m and a radius of $$2$$ m. Find the rate at which the water is leaking out of the cylinder if the rate at which the height is decreasing is $$10$$ cm/min when the height is $$1$$ m.

The water flows out at rate $$\frac{2π}{5}\,\text{m}^3\text{/min}$$.

30) A trough has ends shaped like isosceles triangles, with width $$3$$ m and height $$4$$ m, and the trough is $$10$$ m long. Water is being pumped into the trough at a rate of $$5\,\text{m}^3\text{/min}$$. At what rate does the height of the water change when the water is $$1$$ m deep?

31) A tank is shaped like an upside-down square pyramid, with base of $$4$$ m by $$4$$ m and a height of $$12$$ m (see the following figure). How fast does the height increase when the water is $$2$$ m deep if water is being pumped in at a rate of $$\frac{2}{3} \text{ m}^3$$/sec?

$$\frac{3}{2}$$ m/sec

For exercises 32 - 34, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of $$25 \,\text{ft}^3$$/min. The radius of the pool is $$10$$ ft.

32) Find the rate at which the depth of the water is changing when the water has a depth of $$5$$ ft.

33) Find the rate at which the depth of the water is changing when the water has a depth of $$1$$ ft.

$$\frac{25}{19π}$$ ft/min

34) If the height is increasing at a rate of $$1$$ in/sec when the depth of the water is $$2$$ ft, find the rate at which water is being pumped in.

35) Gravel is being unloaded from a truck and falls into a pile shaped like a cone at a rate of $$10 \,\text{ft}^3/\text{min}$$. The radius of the cone base is three times the height of the cone. Find the rate at which the height of the gravel changes when the pile has a height of $$5$$ ft.

$$\frac{2}{45π}$$ ft/min

36) Using a similar setup from the preceding problem, find the rate at which the gravel is being unloaded if the pile is $$5$$ ft high and the height is increasing at a rate of $$4$$ in/min.

In exercises 37 - 41, draw the situations and solve the related-rate problems.

37) You are stationary on the ground and are watching a bird fly horizontally at a rate of $$10$$ m/sec. The bird is located $$40$$ m above your head. How fast does the angle of elevation change when the horizontal distance between you and the bird is $$9$$ m?

The angle decreases at $$\frac{400}{1681}$$ rad/sec.

38) You stand $$40$$ ft from a bottle rocket on the ground and watch as it takes off vertically into the air at a rate of $$20$$ ft/sec. Find the rate at which the angle of elevation changes when the rocket is $$30$$ ft in the air.

39) A lighthouse, $$L$$, is on an island $$4$$ mi away from the closest point, $$P$$, on the beach (see the following image). If the lighthouse light rotates clockwise at a constant rate of $$10$$ revolutions/min, how fast does the beam of light move across the beach $$2$$ mi away from the closest point on the beach?

$$100π$$ mi/min

40) Using the same setup as the previous problem, determine at what rate the beam of light moves across the beach $$1$$ mi away from the closest point on the beach.

41) You are walking to a bus stop at a right-angle corner. You move north at a rate of $$2$$ m/sec and are $$20$$ m south of the intersection. The bus travels west at a rate of $$10$$ m/sec away from the intersection – you have missed the bus! What is the rate at which the angle between you and the bus is changing when you are $$20$$ m south of the intersection and the bus is $$10$$ m west of the intersection?

The angle is changing at a rate of $$\frac{11}{25}$$ rad/sec.

In exercises 42 - 45, refer to the figure of baseball diamond, which has sides of 90 ft.

42) [T] A batter hits a ball toward third base at $$75$$ ft/sec and runs toward first base at a rate of $$24$$ ft/sec. At what rate does the distance between the ball and the batter change when $$2$$ sec have passed?

43) [T] A batter hits a ball toward second base at $$80$$ ft/sec and runs toward first base at a rate of $$30$$ ft/sec. At what rate does the distance between the ball and the batter change when the runner has covered one-third of the distance to first base? (Hint: Recall the law of cosines.)

The distance is increasing at a rate of $$62.50$$ ft/sec.
44) [T] A batter hits the ball and runs toward first base at a speed of $$22$$ ft/sec. At what rate does the distance between the runner and second base change when the runner has run $$30$$ ft?
45) [T] Runners start at first and second base. When the baseball is hit, the runner at first base runs at a speed of $$18$$ ft/sec toward second base and the runner at second base runs at a speed of $$20$$ ft/sec toward third base. How fast is the distance between runners changing 1 sec after the ball is hit?
The distance is decreasing at a rate of $$11.99$$ ft/sec.