Turvey for Related Rates - waynesville.k12.mo.us

turvey for related rates (key)

I didn’t have time to find my “notebook” lesson with the answers
worked out so I’m just posting this typed key – we will catch the

rest when you get back

In 1953, Roger Price invented a minor art form called the Droodle,

which he described as "a borkley-looking sort of drawing that doesn't

make any sense until you know the correct title." In 1985, Games

Magazine took the Droodle one step further and created the Turvey.

Turvies have one explanation right-side-up and an entirely different
one turned topsy-turvey (when standing on your head – or turning the

paper upside down, if you prefer).
Here is the title right-side-up:

"A C L O S E - U P O F D R A C U L A ."
47 5 2
52 6 3 13

Here is the title upside-down:

“    T W O S H A R K S N O T O N S P E A K I N G

8 12 6 11 7 9 6 10 8 10 6 3 9 10 1

T E R M S ."
8 7 14 6

To determine the titles to this turvey, solve the 14 Related Rates problems. Then
replace each numbered blank with the letter corresponding to the answer for that
problem. The unnumbered blanks are all vowels. You should be able to determine
both titles.

1-2. If a snowball is melting at the rate of 10 ft3 / min, at what rate is the radius
changing when the snowball is 1 foot in radius (Problem #1)? At what rate is the
radius changing when the snowball is 2 feet in radius (Problem #2)?

V  4  r 3
3

dV  4 r2 dr  10  4 22 dr
dt dt dt

 10  4 12 dr dr  10  5 ft / min #2 L
dt dt 16 8

dr  5 ft / min #1 G
dt 2

3-4. A baseball diamond is a square with sides 90 feet. Derek Jeter runs from first base
to second base at 25 feet per second. How fast is he moving away from home plate
when he is 30 feet from first base (Problem #3)? How fast is he moving away from
home plate when he is 45 feet from first base (Problem #4)?

x2  902  y2

2x dx  2 y dy (45)(25)  10125 dy
dt dt dt

(30)(25)  900 dy dy  5 5 ft / s # 4 D
dt dt

dy  5 10 ft / s #3 P
dt 2

5. Water flows at 8 cubic feet per minute into a cylinder with radius 4 feet. How fast
is the water level rising when the water is 2 feet high?

V   (4)2 h

dV  16 dh
dt dt

8  16 dh
dt

dh  1 ft / min #5C
dt 2

6-7. An inverted cone with height 20 meters and radius 5 meters is being filled with a
hose which pumps in water at the rate of 3 cubic meters per minute. When the
water level is 2 meters, how fast is the water level rising (Problem #6)? How fast is

the radius changing at this moment (Problem #7)?

V  1  r 2 h r  5  1 r  h h  4r
3 h 20 4 4
1
1 ( h2 V  3  r 2 (4r )
3 16
V   )h

1 V  4  r 3 r  1
48 3 h 4
V   h3

dV  1  h2 dh dV  4 r2 dr r  1
dt 16 dt dt dt 2 4

3  1  22 dh 3  4 1 dr 1
16 dt 4 dt r2

dh  12 m / min # 6 S dr  3 m / min #7 R
dt  dt 

8-9. A stone is dropped into Burke Lake, causing circular ripples whose radii increase
by 2 m/s. How fast is the disturbed area growing when the outer ripple has

radius 5 meters (Problem #8)? How fast is the radius increasing at that time(#9)?

A  r2 dr  2 m / s #9 K
dt
dA  2r dr
dt dt #8T
dA  2 (5)(2)
dt
dA  20 m2 / s
dt

10-11. A fish is being reeled in at a rate of 2 ft / s by a fisherman at Burke Lake.
If the fisherman is sitting on the dock 30 feet above the water, how

fast is the fish moving through the water when the line is 50 feet long

(#10)? How fast is the fish moving when the line is only 31 feet (#11)?

302  x2  z2

2x dx  2z dz 61 dx  31(2)
dt dt dt

40 dx  (50)(2) dx  62 ft / s #11 H
dt dt 61

dx  5 ft /s #10 N
dt 2

12. A student at Bruin High was painting the school and standing at the top of a 25-
foot ladder. He was horrified to discover that the ladder began sliding away from the
base of the school at a constant rate of 2 ft/ s. At what rate was the top of the ladder
carrying him toward the ground when the base of the ladder was 17 feet away from
the school?

x2  y2  252

2x dx  2 y dy  0 #12 W
dt dt

(17)(2)  336 dy  0
dt

dy  34  1.855 ft / s
dt 336

13. A spherical balloon was losing air at the rate of 5 in3/s. At what rate is the
radius of the balloon decreasing when the radius equals 5 inches?

V  4  r 3
3

dV  4 r2 dr
dt dt

5  4 (25) dr
dt

dr  1 in / s #13 F
dt 20

14. Oil spills into the Potomac in a circular pattern. If the radius of the circle
increases at a constant rate of 3 ft/min, how fast is the area of the spill
increasing at the end of 10 minutes?

A  r2

dA  2r dr #14 M
dt dt
dA  2 (3x30)(3)
dt
dA  180 ft2 / min
dt

Answers: C D FGHKL
B 1
O 2 55  1  5 62 61 2  5
20 2 61 8

MN P R S T VW
180 12 20 -17 336 -1.855
5 5 10 3  336

22 