Exercise on related rates and implicit differentiation

1. Each edge of a cube is expanding at the rate of 1 centimeter (cm) per second. How fast is the volume changing when the length of each edge is (a) 5 cm? (b) 10 cm? (c) $x$ cm?

2. An airplane flies in level flight at constant velocity, eight miles above the ground. (In this exercise assume the earth is flat.) The flight path passes directly over a point P on the ground. The distance from the plane to P is decreasing at the rate of 4 miles per minute at the instant when this distance is 10 miles. Compute the velocity of the plane in miles per hour.

3. A baseball diamond is a 90-foot square. A ball is batted along the third-base line at a constant speed of 100 feet per second. How fast is its distance from first base changing when (a) it is halfway to third base? (b) it reaches third base?

4. A boat sails parallel to a straight beach at a constant speed of $12$ miles per hour, staying $4$ miles offshore. How fast is it approaching a lighthouse on the shoreline at the instant it is exactly $5$ miles from the lighthouse ?

5. A reservoir has the shape of a right-circular cone. The altitude is 10 feet, and the radius of the base is 4 ft. Water is poured into the reservoir at a constant rate of 5 cubic feet per minute. How fast is the water level rising when the depth of the water is 5 feet if (a) the vertex of the cone is up? (b) the vertex of the cone is down?

6. A water tank has the shape of a right-circular cone with its vertex down. Its altitude is 10 feet and the radius of the base is 15 feet. Water leaks out of the bottom at a constant rate of 1 cubic foot per second. Water is poured into the tank at a constant rate of $c$ cubic feet per second. Compute $c$ so that the water level will be rising at the rate of 4 feet per second at the instant when the water is 2 feet deep.

7. Water flows into a hemispherical tank of radius $10$ feet (flat side up). At any instant, let $h$ denote the depth of the water, measured from the bottom, $r$ the radius of the surface of the water, and $V$ the volume of the water in the tank. Compute $dV/dh$ at the instant when $h=5$ feet. If the water flows in at a constant rate of $5\sqrt{3}$ cubic feet per second, compute $dr/dt$, the rate at which $r$ is changing, at the instant $t$ when $h=5$ feet.

8. A variable right triangle $ABC$ in the $xy$-plane has its right angle at vertex $B$, a fixed vertex $A$ at the origin, and the third vertex $C$ restricted to lie on the parabola MATH The point $B$ starts at the point $(0,1)$ at time $t=0$ and moves upward along the $y$-axis at a constant velocity of $2$ cm/sec. How fast is the area of the triangle increasing when $t=7/2$ sec?

9. The radius of a right-circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as the radius. When the radius is 1 foot the altitude is 6 feet. When the radius is 6 feet, the volume is increasing at a rate of 1 cubic foot per second. When the radius is 36 feet, the volume is increasing at a rate of $n$ cubic feet per second, where n is an integer. Compute $n$.

10. A particle is constrained to move along a parabola whose equation is $y=x^2$. (a) At what point on the curve are the abscissa and the ordinate changing at the same rate? (b) Find this rate if the motion is such that at time $t$ we have $x=\sin t$ and $y=\sin ^2t$.

11. The equation $x^3+y^3=1$ defines $y$ as one or more functions of $x$. (a) Assuming the derivative $y^{\prime }$ exists, and without attempting to solve for $y$, show that $y^{\prime }$ satisfies the equation MATH. (b) Assuming the second derivative MATH exists, show that MATH whenever $y\ne 0$.

12. If $0<x<5$, the equation $x^{1/2}+y^{1/2}=5$ defines $y$ as a function of $x$. Without solving for $y$, show that the derivative $y^{\prime }$ has a fixed sign. (You may assume the existence of $y^{\prime }$.)

13. The equation $3x^2+4y^2=12$ defines $y$ implicitly as two functions of $x $ if MATH. Assuming the second derivative $y^{\prime \prime }$ exists, show that it satisfies the equation MATH.

14. The equation $x\sin xy+2x^2{}=0$ defines $y$ implicitly as a function of $x$. Assuming the derivative $y^{\prime }$ exists, show that it satisfies the equation MATH

15. If $y=x^{r}$, where $r$ is a rational number, say $r=m/n$, then $y^{n}=x^{m}$. Assuming the existence of the derivative $y^{\prime }$, derive the formula MATH using implicit differentiation and the corresponding formula for integer exponents.

Exercise on Related Rates

1. The sides of the rectangle increase in such a way that $dz/dt=1$ and $dx/dt=3dy/dt.$ At the instant when $x=4$ and $y=3$, what is the value of $dx/dt?$
MATH

2. A person 2 meters tall walks directly away from a street light that is 8 meters above the ground. If the person's shadow is lengthening at the rate of 4/9 meters per second, at what rate in meters per second is the person walking?
MATH

3. An observer at point A is watching balloon B as it rises from point C. The balloon is rising at a constant rate of 3 meters per second (this means dy/dt = 3) and the observer is 100 meters from point C.
MATH

(a) Find the rate of change in z at the instant when y = 50. (They want dz/dt.)

(b) Find the rate of change in the area of right triangle BCA when y = 50.

(c) Find the rate of change in $\theta $ when y = 50. (They want $d\theta /dr.$)

4. (Calculus classic) The bottom of a 10-foot ladder is going away from the wall at dx/dt = 2 feet per second. How fast is the top going down the wall? Draw the right triangle to find dy/dt when the height y is (a) 6 feet (b) 5 feet (c) zero.

5. The top of the 10-foot ladder can go faster than light. At what height y does dy/dt = c?

6. How fast does the level of a Coke go down if you drink a cubic inch a second? The cup is a cylinder of radius 2 inches first write down the volume.

7. A jet flies at 8 miles up and 560 miles per hour. How fast is it approaching you when (a) it is 16 miles from you; (b) its shadow is 8 miles from you (the sun is overhead); (c) the plane is 8 miles from you (exactly above)?

8. Starting from a 3-4-5 right triangle, the short sides increase by 2 meters/second but the angle between them decreases by 1 radian/second. How fast does the area increase or decrease?

9. A pass receiver is at x = 4, y = 8t. The ball thrown at t = 3 is at x = c(t - 3), y = 10c(t - 3).

(a) Choose c so the ball meets the receiver.

(b) At that instant the distance D between them is changing at what rate?

10. A thief is 10 meters away (8 meters ahead of you, across a street 6 meters wide). The thief runs on that side at 7 meters/ second, you run at 9 meters/second. How fast are you approaching if(a) you follow on your side; (b) you run toward the thief; (c) you run away on your side?

11. A spherical raindrop evaporates at a rate equal to twice its surface area. Find dr/dr.

12. Starting from P= V= 5 and maintaining PV= T, find dV/dt if dP/dt = 2 and dT/dt = 3.

13. (a)The crankshaft AB turns twice a second so d$\theta $/dt =_____.

(b) Differentiate the cosine law 6$^{2}$ = 3$^{2}$ + x$^{2}$ - 2 (3x cos $\theta $) to find the piston speed dx/dt when $\theta =\pi /2$ and $\theta =\pi .$
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14. A camera turns at C to follow a rocket at R.

(a) Relate dz/dt to dy/dt when y = 10. (b) Relate d$\theta $/dr to dy/dt based on y = 10 tan $\theta $. (c) Relate d$^{2}\theta $/dt$^{2}$ to d$^{2}$y/dt$^{2}$ and dy/dt.
MATH

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