Homework #3 for Math Challengers Team Training
Name : _________________________________
1. An equilateral triangular region of side length 2 cm is cut from a corner of an equilateral
triangular region with side length 6 cm. What is the number of centimeters in the
perimeter of the resulting trapezoid?
An equilateral triangular region of side length 2 cm is cut from a corner
of an equilateral triangular region with side length 6 cm as in the
following picture.
The resulting figure shows the smaller triangle with sides of 2 cm and a trapezoid with
sides 2, 4, 6, and 4 cm, respectively. The perimeter of the trapezoid is 2 + 4 + 6 + 4 =
16.
2. An investment of $10,000 is made in a government bond that will pay 6% interest
compounded annually. At the end of five years, what is the total number of dollars in this
investment? Express your answer to the nearest whole number.
An investment of $10,000 is made that pays 6% interest compounded annually. At the
end of 1 year there will be a total of 106% of the original value, i.e., multiply by 1.06.
10000 × 1.06 = 10600
At the end of 2 years there is: 10600 × 1.06 = 11236
At the end of 3 years there is: 11236 × 1.06 = 11910.16
At the end of 4 years there is: 11910.16 × 1.06 = 12624.7696
At the end of 5 years there is: 12624.7696 × 1.06 = 13382.25578
Rounding to the nearest whole number, there are $13382 at the end of 5 years.
3. Washington, D.C. has a land area of 68 square miles and a population of 570,000.
What is the number of people per square mile in Washington, D.C., rounded to the
nearest hundred?
Washington, D.C. has a land area of 68 square miles and a population of 570,000. To
find the number of people per square mile, we must divide the population by the land
area.
570000÷68 = 8382.352941
Rounding to the nearest hundred, there are approximately 8400 people per square mile.
4. A large cube is constructed from individual unit cubes and then dipped into paint.
Thereafter, it is disassembled into the original unit cubes; 486 of these have exactly one
face painted. How many unit cubes were used to construct the large cube?
Homework #3 for Math Challengers Team Training
When a cube, constructed from individual unit cubes, is painted, only those cubes that
do not touch an edge and are on the outside of the cube have exactly one face painted.
Let’s look at cubes with sides of 5 and 4 units, respectively.
Each face of a cube with sides of length 5 has a 3 (i.e., 5 - 2) by 3 square which
contains unit cubes that have only one side painted. Each face of a cube with sides of
length 4 has a 2 (i.e., 4 - 2) by 2 square that contains unit cubes with only one side
painted. This will be true for all 6 faces of the cube. Therefore, there are 6 × (n - 2)2 unit
cubes that have one side painted in a cube whose side is n units.
486 = 6 × (n - 2)2
81 = (n -2)2
92 = (n-2)2
n must be positive.
9=n-2
n = 11
The number of unit cubes is equal to the volume of the cube. 11 × 11 × 11 = 1331
5. The Pittsburgh Stadium Authority passed a resolution that there must be one restroom
for every 70 women attending a game. At a sellout game, at most 42% of the 52,000
attendees are women. How many restrooms are necessary?
42% of the 52,000 attendees at a game are women. Therefore, 52000 × 0.42 = 21840
are women. There must be one restroom for every 70 women.
21840 ÷ 70 = 312
6. How many of the natural numbers from 1 to 600, inclusive, contain the digit 5 at least
once? (The numbers 152 and 553 are two natural numbers that contain the digit 5 at
least once, but 430 is not.)
We must determine how many of the natural numbers from 1 to 600, inclusive, contain
the digit 5 at least once. First look at the numbers from 1 to 99. The numbers that
contain 5 are 5, 15, 25, 35, 45, 50-59, 65, 75, 85, and 95. Thus, there are 19 numbers.
(You can look at it as 10 numbers that start with 5 and 10 numbers than end in 5. But
remember that 55 is counted twice, so 10 + 10 - 1 = 19.) The same holds true for 100 to
199, 200 to 299, 300 to 399, and 400 to 499. However, for 500 to 599, all numbers start
with 5 so all 100 have to be counted. 600 doesn’t have any 5’s. Thus, we have 19 + 19
+ 19 + 19 + 19 + 100 = 19 × 5 + 100 = 195
Homework #3 for Math Challengers Team Training
7. A ferris wheel with radius 90 feet makes a complete circuit every 7 minutes. To the
nearest tenth, how fast is a person in a seat traveling in feet per minute?
A ferris wheel has a radius of 90 feet. The entire circuit is its circumference or 180π . If it
takes 7 minutes to make the complete circuit, then we must divide the circumference by
7 to determine the rate in feet per minute.
180𝜋
180 × 3.141592654
=
= 80.78381109
7
7
Rounding to the nearest tenth, 80.78381109 ≈ 80.8
8. At the Word Store, each vowel sells for a different price, but all consonants are free.
The word “triangle” sells for $6, “ ”square”” sells for $9, “ ”pentagon”” sells for
$7, ”cube”” sells for $7 and “ ”tetrahedron”” sells for $8. What is the dollar cost of the
word “ ”octahedron”” ?
The word triangle contains 1 i, 1 a and 1 e and sells for $6. The word square contains 1
u, 1 a and 1 e and sells for $9. The word pentagon contains 1 e, 1a and 1 o and sells for
$7. The word cube contains 1 u and 1 e and sells for $7. The word tetrahedron contains
2 e’s, 1 a and 1 o and sells for $8. The word octahedron contains 2 o’s, 1 a and 1 e.
Writing these as equations:
i + a + e = 6 (Eq. 1)
u + a + e = 9 (Eq. 2)
e + a + o = 7 (Eq. 3)
u + e = 7 (Eq. 4)
2e + a + o = 8 (Eq. 5)
We must determine the value of 2o + a + e.
e = 1 (Eq. 5 - Eq. 3)
u + 1 = 7 (from Eq. 4)
u=6
6 + a + 1 = 9 (from Eq. 2)
a=2
1 + 2 + o = 7 (from Eq. 3)
o=4
2o + a + e = 2 × 4 + 2 + 1
= 8 + 3 = 11