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Mu Alpha Theta 2004 State Bowl Questions - Mu
Round 1
1. Find all value(s) of t , such that tt b  1067 .
2. The length of a rectangle is increasing at 3 times the rate that its width is increasing.
The diagonal of the rectangle is increasing how many times faster than the width when
the length is 12 feet and the width is 6 feet?
Round 2
3. Let f ( x ) be a linear function such that
2
0
f(x)dx  10 . Find the sum of the slope and the
y-intercept of the graph of f.
4. Find the average value of f (x) = sin x on [ 0,

].
2
Round 3
5.
 x x  3dx 
 ax 3  6x; if x  1

then a 
6. If the function f x is continuous and differentiable and f x =  2
bx  4; if x>1



Round 4
7. Let P = number of grams of the radioactive element petetonium. The rate of change in P is
directly proportional to the amount present. At t = 1, there are 16 grams of P present. At
t = 5, there are 4 grams present. At what time t will there be exactly 1 gram of P present?
x2
y2

 1 with sides parallel to the
100 50
coordinate axes. What is the maximum area of the rectangle?
8. A rectangle is to be inscribed in the ellipse
Round 5
9. If y



dy
 4  y 2 x 2 , and y 0  12, find y.
dx
10. If cos (9x) – cos (7x) = 0, find the number of solutions for x, where 0 < x <


.
2
Round 6

11. The radius of convergence of

n1
an
x  2
n
; a  0 is what?
12. Consider the set of all possible four-digit numbers consisting only of the digits 3, 4, 5, 6,
and 7 with no repetition of the digits in a given number. Find the sum of all such
numbers.
Round 7
13. Find lim x
1
x
x
3

1 2
14. Find the length of the arc of y  2  x   from x  4 to x  16.
9

Round 8
15. Find the area enclosed by the set of points defined by | x | + | y | + | x + y | = 2.
16. A slice of pie in the form of a sector of a circle is to have a perimeter of 18 inches.
What should be the diameter of the circle in order to make the piece of pie largest?
Round 9
e
17. Evaluate
 lnx  dx.
2
1

3

18. Given that g x is an even function and that  g x dx  5, evaluate
0
 4g x  5dx.
3
3
Round 10
sin y  tan y
y0
3y 3
19. Evaluate lim

20. Find the value of b, greater than 1, such that the average value of f x  x 2 on the
interval 1,b  equals 7.
Mu Alpha Theta National Convention 2004
Mu State Bowl Answers
#
Answer
#
Answer
1-1
1, 5
7-13
1
1-2
7 5
5
7-14
112
2-3
5
8-15
3
8-16
9
9-17
e-2
2
2-4

x  3

5
2
3-5

5
2

3
2 x3
2
C
3-6
-14
9-18
70
4-7
9
10-19
1
6
4-8
100 2
10-20
4
5-9
148e
2x 3
3
4
5-10
4
6-11
x  a  2 or x  -a - 2
6-12
666,600
1. Find all value(s) of t , such that tt b  1067
Answer: 1, 5
Solution: 106 7  55 . Hence, tb  t  t(b 1)  55 . Since t must be an integer, can
equal
1, 5, or 11. But it must also be a digit so t  5 , 1 are the only solutions.
2. The length of a rectangle is increasing at 3 times the rate that its width is increasing. The
diagonal of the rectangle is increasing how many times faster than the width when the length
is 12 feet and the width is 6 feet?
7 5
5
Answer:
Solution:
d 2  w2  l 2
180
d 2 =36+144 
d  180
 dw 
dd
dw
dw
 6
 12 3    42
dt
dt
dt
 dt 

  
 
3. Let f(x) be a linear function such that
2d

2
0
dd
dw
dl
 2w
 2l
dt
dt
dt

d
dd
dw
dl
=w
l
dt
dt
dt
dd
42 dw 42 dw 7 dw 7 5 dw




dt
5 dt
180 dt 6 5 dt
5 dt
f(x)dx  10 . Find the sum of the slope and the
y-intercept of the graph of f(x).
Answer: 5
2
mx 2
 bx]20  2m  2b 2m + 2b = 10 so m + b = 5
2
0

4. Find the average value of f ( x ) = sin x on [ 0, ].
2
Solution: Let f(x) = mx + b
 (mx  b)dx 

2
Ans:
 sin x dx
2

Solution:
0

2
5.
x
0


 cos x ] 2

0
2

(0  1)


2

2
x  3dx 
Answer:




5
3
2
x3 2 2 x3 2 C
5
 x  x  3dx
Solution: Using u-substitution, let u  x  3, then du  dx and x  u  3
1
3
5
3
 3

2 52
2
2
2
2
2
2
u  3 u du    u  3u  du  u  2u  C  x  3  2 x  3  C
5
5




1
2




3

 ax  6x; if x  1
then a 
6. If the function f x is continuous and differentiable and f x =  2
bx  4; if x>1

Answer: -14


Solution: To be both differentiable and continuous, the functions and the derivatives must
be equal at x  1. a  6  b  4 and 3ax 2  6  2bx  3a  6  2b  3a  2b  6
b  a  10



3a  2 a  10  6

a  14
7. Let P = number of grams of the radioactive element petetonium. The rate of change in
P is directly proportional to the amount present. At t = 1, there are 16 grams of P
present.
At t = 5, there are 4 grams present. At what time t will there be exactly 1 gram of P
present?
Answer: 9
p'  kp  p  p0 e kt
Solution:
add-multiply property of exponential functions.
t
P(t)
1
16 add four to t and divide P(t) by 4.
5
4
9
1
x2
y2

 1 with sides parallel to the
8. A rectangle is to be inscribed in the ellipse
100 50
coordinate axes. What is the maximum area of the rectangle?
Answer: 100 2
Solution: Solving for y in the ellipse we get y 
2
100  x 2 . The area of the rectangle =
2


1
2 2
2
2 2
(2x)(2 y)  A  2x
100  x  A  2 2x 100  x

2
x 2
x 2  100  x 2
A  2 2
 2 2 100  x 2  0 
 0  2x 2  100  x 2  50
2
2
100  x
100  x
 
x  50  5 2
9. If y
y

 
2
50  5  A  10 10 2  100 2
2


dy
 4  y 2 x 2 , and y 0  12, find y 2 .
dx
Answer: 148e
2x3
3
4
ydy

Solution: By separation of variables 
4  y2

x 2 dx 


1
x3
ln 4  y 2 
C
2
3
2x
2x
 4  y 2  2x 3
1
2x 3
4  y2
2
3
3
ln 148  C  ln 4  y 2 
 ln148  ln 



e

y

148e
4

2
3
3
148
 148 
 


3
3
10. If cos (9x) – cos (7x) = 0, find the number of solutions for x, where 0 < x <

.
2
Ans: 4
Solution: cos (9x) – cos (7x) = 0  cos (8x + x) – cos (8x – x) = 0
cos 8x cos x – sin 8x sin x – [cos 8x xos x + sin 8x sin x] = 0

  3 
,
-2sin 8x sin x = 0  sin 8x sin x = 0 8x = , 2, 3, 4 x = , ,
8 4 8 2

an
; a  0 is what?
11. The radius of convergence of 
n
n1 x  2

Answer: x  a  2 or x  -a - 2


n
Solution:
 a 
The infinite series of 
will converge when
 x  2 
a
 1 means
x2
x2
a
1
 1  x+2 >a  x  2  a or x  2  a  x  a  2 or x  -a - 2
a
x2
12. Consider the set of all possible four-digit numbers consisting only of the digits 3, 4, 5, 6,
and 7 with no repetition of the digits in a given number. Find the sum of all such
numbers.
Ans: 666,600
Solution: There are 120 possible four-digit numbers. 5 is the average of the digits therefore
the average of the four-digit numbers is 5555. 120 x 5555 = 666,600
1
13.
Find lim x x
x
Answer: 1
Solution:
For this limit of indeterminate form, we need to get it into the
1
L’Hopital’s rule y  x x  ln y 
1
ln x
lim
 lim x  0  ln y  0
x x
x 1
  form to use
g x 
f x
1
ln x
ln x 
 now L'Hopital's rule can be applied
x
x

y  e0  1
3

1 2
14. Find the length of the arc of y  2  x   from x  4 to x  16.
9

Answer: 112
Solution:
1

dy
1 2
 3 x  
dx
9

16

4
2
 dy 

1
 dx   9  x  9   9x  1

 2  23
1  9x  1 dx  3 x dx  3  x
 3


16
1
2
4
16
4

 2  16


   4   2 64  8 2  56  112
3
2
3
2
15. Find the area enclosed by the set of points defined by | x | + | y | + | x + y | = 2.
Ans: 3
Solution: Area = 1 + .5 + .5 + 1 = 3
(1, 0)
(-1, 0)
16. A slice of pie in the form of a sector of a circle is to have a perimeter of 18 inches.
What should be the diameter of the circle in order to make the piece of pie largest?
Answer: 9
18  2r

 A
r2 
r
2
9
A  9r  r 2  A  9  2r  0  r 
diameter  9
2
Perimeter of a sector = r  r  r  18   
Solution:
1  18  2r  2
r
2  r 
e
17. Evaluate
 lnx  dx.
1
Answer: e  2
Solution:
2
 
Using integration by parts, let u  ln x
e
 
 
2
ln x dx  x ln x
2
1

 1
and let dv  dx. Then du  2 ln x   dx and v  x.
 x
 
e
 1
 2  x ln x   dx  x ln x
1
 x
1
 
e
again with u  ln x and dv  dx

2
 

giving
e
2
1
e
 2  ln xdx
Using integration by parts
1
 
= x lnx
2
e
1

 2 x ln x

e
1
 
 1
 2  x   dx 
 x
1
e
e 1  0  2e  0  2 e  1  e  2
3


18. Given that g x is an even function and that  g x dx  5, evaluate
0
 4g x  5dx.
3
3
Answer: 70
0


Since g x is even,  g x dx  5,
Solution:
-3
0

3
3
3
-3
3
3
      
3

 4g x  5dx  4  g x dx  5  dx 
3
4  g x dx  4  g x dx   5dx  4 5  4 5  5 3  3  20  20  30  70
3
3
0
sin y  tan y
y0
3y 3
19. Evaluate lim
Answer:
1
6
 sin y  2sec 2 y tan y

y0
18y
Solution: Using L’Hopital’s rule 3 times: lim
 cos y  2sec 4 y  4sec 2 y tan 2 y 1 2
1


y0
18
18
6
20. Find the value of b, greater than 1, such that the average value of f x  x 2 on the
lim

interval 1,b  equals 7.
Answer: 4
Solution: f av 
b
1
1  x3 
2
x
dx

b  1 1
b  1  3 



b
1

b3  1
b2  b  1

 7 b2  b  1  21
3
3 b1
 
b2  b  20  0  b  5 b  4  0  b  5 and
b4
only b  4 satisfies the conditions