Download WYSE – “Academic Challenge”

WYSE – “Academic Challenge”
Sectional Mathematics Solution Set – 2003
1. Correct Answer: D.
First add the two vectors together to get –2i + 5j – k. Then find the length of this vector
using the standard method of
(−2) 2 + (5) 2 + (−1) 2 , which gives a result of
30 .
2. Correct Answer: B.
Using trigonometry, part of the statues height can be found using tan31 = x/75 where x is
the height from the top of the statue to five feet above the ground. s = tan31/75 = 45.064. If
you add in the height of the instrument of 5 feet you get a total height of 50 feet.
3. Correct Answer: B.
Begin by letting x = distance the fence is from the pool and recalling P=2W+2L
128 = 2(18 + 2 x) + 2(22 + 2 x) Þ 128 = 36 + 4 x + 44 + 4 x Þ 8 x = 48 Þ x = 6
4. Correct Answer: E.
Look for the only one that corresponds to the known identities of trigonometry.
5. Correct Answer: C.
The sum of the measures of a regular heptagon is found by the formula
(n - 2)180 where n is the number of sides. (7 - 2) 180 = 900. The measure of one angle is
900/7 = 128.57. Dividing this number in two would give you the measure of the base angles
of the isosceles triangles made from connecting consecutive vertices of the heptagon with
its center. So the base angle of the triangle would be 128.57/2 = 64.3. Since the apothem
is 5, then the measure length of one side of heptagon can be found using the following
trigonometry relationship, tan 64.3 = 5/x where x is the length of one half of a side of the
heptagon. Solving this relationship gives you a value of x = 2.41 so that one side length
would be 4.82 resulting in a total perimeter of approximately 33.7
6. Correct Answer: D.
The bottom factors to (x - 3)(x+2) so x can't be 3 or -2. The top also imposes the restriction
that x must be at least 2.
7. Correct Answer: B.
Use either a calculator or standards calculating procedures to find the values. Remember
that this is population standard deviation, so calculating the mean does not cause a loss of a
degree of freedom.
8. Correct Answer: A.
Since BX is a perpendicular bisector or AD, then AB = BD by the perpendicular bisector
theorem. Since BY is the perpendicular bisector of CD, BD = BC. Therefore by the
transitive property, AB = BC.
Sectional Mathematics Solution Set – 2003
9. Correct Answer: A.
This can be written as (3x-2)(5x-4)=0 so x = 2/3 or 4/5 answer a
10. Correct Answer: D.
This situation is directly modeled using the permutation P (10,4).
11. Correct Answer: D.
The function f(x) = -5 sin(2x - 6) + 4 is the same as f(x) = -5 sin 2(x - 3) + 4. The phase
shift is determined from the expression (x - 3) which indicates a horizontal shift of 3 units to
the right.
12. Correct Answer: B.
A distance problem, time our unknown. If we let x = time the sheriff drives
d robbers = d sheriff Þ rrobbers ∗ t robbers = rsheriff ∗ t sheriff Þ 70( x + 1 / 5) = 84( x) Þ 14 x = 14 Þ x = 1
13. Correct Answer: B.
First find the sum of the two vectors, which is 3i + j. Then look for the only vector which
gives a dot product of zero with 3i + j.
14. Correct Answer: B.
First, using pythagorean theorem, you can find out that the hypotenuse of Triangle ABC is
approximately 12.2. Since AX is an altitude, that makes angle AXC a right angle also.
Using properties of similarity between right triangle ABC and XAC you can determine that
the ratio of similitude or scale factor is s = 7/12.2 = .57. To compare areas, however, you
have to square the ratio, .57^2 = .33. In other words, the area of the small triangle is 33% of
the big triangle. Since the area of the big triangle is 1/2(10)(7) = 35, then the area of the
small triangle must be .33(35) = 11.5.
15. Correct Answer: B.
The above equation in y-intercept form is y = (-1/10)x + 1. This line forms a right triangle with
a base of 10 and a height if 1 getting (1/2)(10)(1) = 5
16. Correct Answer: E.
Either use the built in fuction of the inverse function of a calculator, or see which
matrix has rows that are scalar multiples of each other.
Sectional Mathematics Solution Set – 2003
17. Correct Answer: B.
An isometry is any transformation that preserves both side length and angle measure.
While the transformation T(x, y) à (2x, 2y) would preserve angle measure, it increase each
side length by a magnitude of 2 resulting in a dialation of the triangle
18. Correct Answer: C.
A mixture problem. 0.035(3) + 0(x) = 0.02(x + 3) and 0.02x = 0.045 and x = 2.25
19. Correct Answer: A.
Use either a calculator or standards calculating procedures to find the values. Remember
that this is population standard deviation, so calculating the mean does not cause a loss of a
degree of freedom. Due to the large number of units in the population, it’s probably easier
to calculate them using the following method: mean = (50/150)*1 + (80/150)*2 + (20/150)*3
= 1.8, and s.d. =
(50 / 150) * (1 − 1.8) 2 + (80 / 150) * (2 − 1.8) 2 + (20 / 150) * (3 − 1.8) 2 = .653
20. Correct Answer: C.
Let x be the length of one side of the square base. The volume of the prism if found using
V = (Area of the base)(Height of the Prism). Therefore, 105 = (x^2)(12). Solving for x you
get, 2.96. The total surface area is then SA = 2(2.96)(2.96) + 4(2.96)(12) = 159.6
21. Correct Answer: B.
The sum of the numbers from 1 to 321 is 321(322)/2=51,681. We don’t want the sum of the
numbers from 1 to 122, which is 122(123)/2=7,503.
22. Correct Answer: C.
Use basic matrix multiplication to solve for BAC. Be sure to put them in the correct order, as
matrix multiplication does not commute.
23. Correct Answer: B.
The area of the base is 16 sq cm, so the area of the four sides is 56 - 16 = 40 sq cm. The
area of one of the lateral faces is then 40/4 = 10 sq cm. Using the area of 10, you can find
the slant height of pyramid à 10 = 1/2(4)(slant height) à slant height = 5 cm. Using
pythagorean theorem, you can then find the overall height of the pyramid using the slant
height and half of the base length à h^2 + 2^2 = 5^2 à h = 4.58. So the volume is
1/3(16)(4.58) = 24.
24. Correct Answer: C.
300(.5) x / 3.1 = 10 Þ 0.5 x / 3.1 = 1 / 30 Þ
ln(1 / 30)
x
=
Þ x = 15.211
3 .1
ln(.5)
Sectional Mathematics Solution Set – 2003
25. Correct Answer: B.
This problem is best solved with accompanying picture. The current situation can be drawn
with an isoceles triangle with a base being the same segment as the hypotenuse of a right
triangle, with the vertical part of the right triangle on the right side. The hypotenuse has a
length of eight feet and the vertical side of the right triangle has a length of one foot, so the
triangle has a base length of 63 feet. Now, use trigonometry to find the two angles on the
left side of the figure. Because the top triangle is isoceles, it can be split into two right
triangles. This means that the angle in question is cos −1 (4 / 6) , or approximately 48.19
degrees. The angle in question on the lower right triangle is sin −1 (1 / 8) , or approximately
7.18 degrees. This means the two angles together are 55.37 degrees. If we create a
vertical line from the highest point to the base, then a new right triangle is creates, and
solving for sin 55.37 o =
height
will give us how high the highest point is.
6
Answer: B
26. Correct Answer: C.
Rewrite the function in the form f(x) = -3tan (2x - π/2). The locations of the asymptotes are
at (2x - π/2) = π/2 and (2x - π/2) = -π/2. solving for x we get two values, x = π/2 and x = 0.
Therefore the period must be the distance between the asymptotes which is a distance of
π/2.
27. Correct Answer: D.
Using the formula A=P(1+r/n)^nt we get answer D
2 , 500 = 1000 (1 + r / 4 ) 4 * 4 Þ (1 + r / 4 ) 16 = 2 . 5 Þ r / 4 =
16
2 . 5 − 1 Þ r = . 2357
28. Correct Answer: C.
This can either be modeled in two different ways. It can be found by permuting the letters
and then dividing out the duplications for the p’s and e’s by using P (9,9) / (P (3,3) * P (2,2))
= 30240. The number of ways can be also found by placing the e’s, placing the p’s, and
then placing the rest of the letters. This method is modeled by C (9,2) * C (7,3) * P (4,4) =
30240.
29. Correct Answer: A.
The rectangular coordinates are found with the conversion x = 1.5 sin(5) and y = 1.5 cos(5).
Make sure your calculator is set to radians. Therefore the approximate rectangular
coordinates are (-1.44, .43).
30. Correct Answer: B.
Let x = p(any face other than a 6) so 6x = p(a 6) and x + x + x + x + x +6x = 1 and 11x = 1
So x = 1/11. There are 3 primes from 1to 6, namely 2,3 and 5
Sectional Mathematics Solution Set – 2003
31. Correct Answer: C.
There are P (10,3), or 720 possible ways to assign parts. Hildegard wants one of the three,
and then wants the remaining two to be filled by eight of the remaining people. This means
that 3*8*7, or 168 ways satisfy her. This probability she gets what she wants is 168/720, or
.233.
32. Correct Answer: A.
After 3.5 hours the first ship is 45.5 miles away from port and the second ship is 63 mile per
hour away from port. The angle between the two ships is 83 degrees. Using the law of
cosines, we can find the distance between the two with the expression:
d^2 = 45.5^2 - 63^2 - 2(45.5)(63)cos83. So d = 73.
33. Correct Answer: D.
a n = a1 + (n − 1)d Þ 333 = 3 + (n − 1)3 Þ n = 111 and S 111 =
111
(3 + 333) = 18,648
2
34. Correct Answer: B.
é1 0ù
ú when multiplied with
ë0 1û
Look for the matrix that forms ê
éa bù
êc d ú .
ë
û
éa b 1 0ù
ú . Answer: B
ë c d 0 1û
Also possible by reducing ê
35. Correct Answer: A.
Since A // B // C, set up the following proportion using the side splitting theorem:
1 .5
1
x −1
=
Þ x 2 − 1 = 1 . 5 x Þ ( x − 2 )( x + ) = 0
2
x
x +1
This leads to the solutions x=2 and x = -1/2. The context of the problem makes -1/2 an
unusable solution so x in the problem must be 2. Plugging in 2 for x in all the expressions
and adding them up give you a perimeter of 12.5 units.
36. Correct Answer: A.
After re-writing the above equation we get:
x
ln(5)
æ2ö
≈ −3.97
ç ÷ =5Þ x =
ln(2 / 3)
è3ø
Sectional Mathematics Solution Set – 2003
37. Correct Answer: C.
This is a direct application of Baye’s theorem. Of all the people, 3/5*2/3 box fruit pies and
2/5*4/5 box custard pies. This means that out of the 3/5*2/3 + 2/5*4/5 probability a person
boxes a pie, the probability of the pie being a fruit pie is (3/5*2/3) / (3/5*2/3 + 2/5*4/5) =
.555.
38. Correct Answer: C.
To find the perimeter one needs to find the altitude, the length BD and the length AB. The
altitude is found with trigonometry, alt = 130 sin 63.
A
x
y
B
To find AB, AB = Ax + xy + yB. Ax = 130 cos63, xy = 143, and yB = 130 sin63/tan72.
Therefore AB = 59.0 + 143.0 + 37.6 = 239.6. The length from B to D is found by
115.8*sin72 = 121.8. Therefore half of the perimeter is (130 + 143 + 239.6 + 121.8)/2 =
317.2
39. Correct Answer: B.
Getting the equation set up properly is the toughest part for this one. I get:
1æ 1 ö
1
= x Þ 2x 2 − 2x −1 = 0 and x = −0.366 or 1.366
ç
÷=xÞ
2 è x −1 ø
2x − 2
40. Correct Answer: B.
Put both speeds into miles per hour. Car A is traveling at 43 miles per hour, and car B is
traveling at 30.68 miles per hour. Car A takes 10/43 = .233 hours, or 14 minutes to travel
the stretch of road. Car B takes 10/30.68 .326 hours, or 19.6 minutes to travel the same
road. Compare the times.
Sectional Mathematics Solution Set – 2003