# Related Rates Baics

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```					Basics

1. The width of a rectangle is increasing at a rate of 2 cm/sec. and its length is
increasing at a rate of 3 cm/sec. At what rate is the area of the rectangle increasing
when its width is 4 cm and its length is 5 cm?

Ans. 22 cm 2 sec
2. Digging in your backyard, you accidentally break a town water line. Water,
bubbling up at a rate of 1 cubic inch per second, is forming a circular pond of
depth 0.5 inch in your backyard. How quickly is this pond covering your lawn?
(i.e. how quickly is the surface area of this pond increasing?)

Ans. 2in 2 sec
3. A police car, approaching a right-angled intersection from the north is chasing a
speeding car that has turned the corner and is now moving straight east. When the
police car is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the
police determine with the radar that the distance between them and the car is
increasing at 20 mph. If the police car is moving at 60 mph at the instant ofy

measurement, what is the speed of the car?

police car

x

speeding car

Ans. 70 mph
4. A man 6 ft tall is walking away from a spotlight (L) located on the ground. His
shadow is cast on a wall 40 ft from the spotlight. If the man is walking at a rate of
4 ft per second away from the spotlight determine the rate of change of the
shadow (PQ) when he is half way to the wall.                            y

Q

B

6
x

L       A            P

12 ft
Ans.
5 sec

5. Water runs into a conical tank at a the rate of 9 ft 3 / min . The tank stands point
down and has a height of 10 ft and a base radius of 5 ft. How fast is the water
level rising when the water is 6 ft. deep?

1 ft
Ans.
 min

6. The Pillsbury dough-boy has decided to bake cookies for the Calculus classes. He
begins with 40 cm 3 of cookie dough. Rolling the dough into a cylindrical shape
he notices that the rate of change of the radius of the cylinder is decreasing at .5
cm per minute. At the instant when the radius of the cylinder is 2cm determine
the rate of change of the height of the cylinder

Ans. 5cm min
7. In the triangle shown above, if  increases at a constant rate of 3 radians per
minute, at what rate is x increasing in units per minute when x equals 3 units?

5
x



Ans. 12 units min

mi
8. An airplane (pt. A) is flying 600       on a horizontal path that will take it directly
hr
over an observer (pt. O). The airplane maintains a constant altitude of 7 mi (see
figure). What is the rate of change of the distance between the observer and the           A
airplane when x = 5 mi.?

z
7 mi.

O
x

3000
Ans.         mph
74

mi
9. An airplane (pt. A) is flying 600     on a horizontal path that will take it directly
hr
over an observer (pt. O). The airplane maintains a constant altitude of 7 mi (see
figure). How fast is the angle  changing when x = 5 mi.?                             A

z
7 mi.

O
x