Solutions to Set 8

Problem #1: 3
Problem #2: 729/1000
Problem #3: 17
Problem #4: 8
Problem #5: -2
Problem #6: 12
Problem #7: 108
Problem #8: 96
Problem #9: 40
Problem #10: 6

Explanations

Problem #1: If x=1, then
30x + 1 = 30(1) + 1 = 31,
which is prime.
If x=2, then
30x + 1 = 30(2) + 1 = 61,
which is prime.
If x=3, then
30x + 3 = 30(3) + 1 = 91,
which is NOT prime because
91 = 7 * 13.
Therefore, 3 is the smallest value of x fulfilling the desired conditions.
Read more about Prime Numbers.
Problem #2: The ratio of the diameters is
90% = 9/10.
This is the ratio of two lengths in similar shapes, so the ratio of their volumes is found by cubing that ratio.
(9/10)³
= 729/1000.
Problem #3: Let x represent the desired integer. You know that
x < √321
Squaring both sides gives
x² < 321
The largest perfect square less than 321 is 17²=289, so x²=289, and x=17.
Read more about Perfect Squares.
Problem #4: First, think about the perfect squares closest to 70. You know that
64 < 70 < 81.
√64 < √70 < √81
8 < √70 < 9.
But, which is closer: 8 or 9? To find out, calculate 8.5².
8.5² = 72.25.
You can now narrow down the range:
64 < 70 < 72.25
√64 < √70 < √72.25
8 < √70 < 8.5.
Therefore, √70 must be closest to the integer 8.
Read more about Perfect Squares.
Problem #5: Use the formula for the slope of a line, given two points (x₁, y₁) and (x₂, y₂):
slope = (y₂ - y₁)/(x₂ - x₁)
= (0.6 - 10.4)/(12.4 - 7.5)
= -2.
Problem #6: A quick sketch shows that AB is a vertical line segment and that AC is a horizontal line segment. The lengths of these are:
AB = 5 - 2 = 3
AC = 6 - 2 = 4
Since a horizontal line segment and a vertical line segment form a right angle, ABC is a right triangle. Therefore, you can use the Pythagorean theorem or knowledge of the Pythagorean triple (3, 4, 5) to find that
BC = 5.
The perimeter of the triangle, therefore, is
AB + AC + BC
= 3 + 4 + 5
= 12.
Read more about The Pythagorean Theorem.
Problem #7: Start by setting up an equation, using the formula for the number of diagonals in a polygon. Then solve this equation to find out the number of sides in the polygon:
n = number of sides
[n(n - 3)]/2 = n
n(n - 3 = 2n
n - 3 = 2
n = 5.
Then, use the formula for the measure of each interior angle of a regular n-sided polygon:
(n - 2)(180/n)
= (5 - 2)(180/5)
= (3)(36)
= 108 degrees.
Read more about Formulas About Polygons.
Problem #8: Suppose that Izzy's car drove D meters. Then, at 3 meters per second, it took her D/3 seconds to finish the race.
Since Sam got a 16-meter head start, her car only had to drive D-16 meters. Then, at 2.5 meters per second, it took her (D-16)/2.5 seconds to finish the race.
The race ended in a tie, so Izzy's and Sam's cars must have taken the same amount of time to travel their respective distances. Therefore,
D/3 = (D - 16)/2.5
2.5D = 3(D - 16)
2.5D = 3D - 48
48 = .5D
D = 96 meters.
Problem #9: Find the prime factorization of 111.
The divisibility rule for multiples of 3 states, "A number is divisible by 3 if and only if the sum of the digits is divisible by 3." Therefore, 111 is divisible by 3.
By dividing 111 by 3, you find that 111=3*37. Since both 3 and 37 are prime numbers, this is the complete prime factorization.
So, the answer is 3+37=40.
Read more about Prime Factorization and Prime Numbers.
Problem #10: Work backward. Because 33 is the sum of the two previous terms, c and 20,
c = 33 - 20 = 13.
Similarly,
b = 20 - c = 20 - 13 = 7
a = c - b = 13 - 7 = 6.