MATHEMATICS HIGH SCHOOL

An 8-sided fair die is rolled twice and the product of the two numbers obtained when the die is rolled two times is calculated.(a) Draw the possibility diagram of the product of the two numbers appearing on the die in each throw (b) Use the possibility diagram to calculate the probability that the product of the two numbers is I) A prime number ii) Not a perfect square iii) A multiple of 5 iv) Less than or equal to 21 v) Divisible by 4 or 6

Answers

Answer 1

Answer:

Answer:

(a) Shown below.

(b) Explained below.

Step-by-step explanation:

(a)

The sample space of rolling an 8-sided die twice is as follows:

S = {(1 , 1) , ( 1 , 2) , ( 1, 3) , ( 1, 4 ) , ( 1, 5) , ( 1 , 6 ) , ( 1, 7 ) , ( 1, 8) ,

(2 , 1) , (2 , 2) , ( 2, 3) , ( 2, 4 ) , ( 2, 5) , (2 , 6 ) , ( 2, 7 ) , ( 2, 8) ,

(3 , 1) , ( 3, 2) , ( 3, 3) , ( 3, 4 ) , ( 3, 5) , ( 3 , 6 ) , (3, 7 ) , ( 3, 8) ,

(4, 1) , ( 4 , 2) , ( 4, 3) , ( 4, 4 ) , ( 4, 5) , (4 , 6 ) , (4, 7 ) , (4, 8) ,

(5, 1) , ( 5 , 2) , ( 5, 3) , (5, 4 ) , ( 5 ,5) , (5, 6 ) , ( 5, 7 ) , ( 5, 8) ,

(6 , 1) , ( 6 , 2) , ( 6, 3) , (6, 4 ) , ( 6, 5) , (6 , 6 ) , ( 6, 7 ) , ( 6, 8) ,

(7 , 1) , ( 7 , 2) , ( 7, 3) , ( 7, 4 ) , ( 7 , 5) , ( 7, 6 ) , ( 7, 7 ) , (7, 8) ,

(8 , 1) , ( 8 , 2) , (8, 3) , ( 8, 4 ) , ( 8, 5) , ( 8 , 6 ) , ( 8, 7 ) , ( 8, 8)}

There are a total of N = 64 elements.

(b)

(i)

The product of the two numbers is a prime number:

Product is a prime number samples:

2 = ( 1, 2) , ( 2, 1)

3 = ( 1 , 3) , ( 3 , 1)

5 = ( 1, 5) , ( 5 , 1)

7 = ( 1, 7) , ( 7 , 1)

Number of samples, n = 8

P (Product is a prime number) = 8/64 = 1/8 = 0.125.

(ii)

The product of the two numbers is not a perfect square :

Product is not a perfect square samples:

2 = ( 1, 2) , ( 2, 1)

3 = ( 1 , 3) , ( 3 , 1)

5 = ( 1, 5) , ( 5 , 1)

6 = ( 1, 6) , ( 2, 3) , ( 3, 2) , ( 6 , 1)

7 = ( 1, 7) , ( 7 , 1)

8 = ( 1 , 8) , ( 2, 4) , ( 4 , 2) , ( 8 , 1)

10 = ( 2, 5) , ( 5, 2)

12 = (2 , 6) , ( 3 , 4) , ( 4 3) , ( 6 , 2)

14 = ( 2, 7) , ( 7 , 2)

15 = (3 , 5) , ( 5 , 3)

18 = ( 3, 6) , ( 6 , 3)

20 = ( 4, 5) , ( 5, 4)

21 = ( 3 , 7) , ( 7 , 3)

24 = ( 3 , 8) , ( 4 , 6 ) , ( 6 , 4) , ( 8 , 3)

28 = ( 4 , 7) , ( 7 , 4)

30 = ( 5 , 6) , ( 6 ,5 )

32 = ( 4 , 8) , ( 8 , 4)

35 = ( 5 , 7) , ( 7 , 5)

40 = ( 5 , 8) , ( 8 , 5)

42 = ( 6 , 7) , ( 7 , 6)

48 = ( 6 , 8) , ( 8 , 6)

56 = ( 7 , 8) , ( 8 , 8)

Number of samples, n = 52

P (Product is not a perfect square) = 52/64 = 0.8125

(iii)

The product of the two numbers is a multiple of 5:

Product is a multiple of 5 samples:

5 = ( 1, 5) , ( 5 , 1)

10 = ( 2, 5) , ( 5, 2)

15 = (3 , 5) , ( 5 , 3)

20 = ( 4, 5) , ( 5, 4)

25 = ( 5 , 5)

30 = ( 5 , 6) , ( 6 ,5 )

35 = ( 5 , 7) , ( 7 , 5)

40 = ( 5 , 8) , ( 8 , 5)

Number of samples, n = 15

P (Product is a multiple of 5 ) = 15/64 = 0.2344.

(iv)

The product of the two numbers is less than or equal to 21:

Product is less than or equal to 21 samples:

1 = ( 1, 1)

2 = ( 1, 2) , ( 2, 1)

3 = ( 1 , 3) , ( 3 , 1)

4 = (1 , 4) , ( 2, 2) , ( 4, 1)

5 = ( 1, 5) , ( 5 , 1)

6 = ( 1, 6) , ( 2, 3) , ( 3, 2) , ( 6 , 1)

7 = ( 1, 7) , ( 7 , 1)

8 = ( 1 , 8) , ( 2, 4) , ( 4 , 2) , ( 8 , 1)

9 = ( 3, 3)

10 = ( 2, 5) , ( 5, 2)

12 = (2 , 6) , ( 3 , 4) , ( 4 3) , ( 6 , 2)

14 = ( 2, 7) , ( 7 , 2)

15 = (3 , 5) , ( 5 , 3)

16 = (2 , 8) , ( 4 , 4) , ( 8 , 2)

18 = ( 3, 6) , ( 6 , 3)

20 = ( 4, 5) , ( 5, 4)

21 = ( 3 , 7) , ( 7 , 3)

Number of samples, n = 40

P (Product is less than or equal to 21) = 40/64 = 0.625.

(v)

The product of the two numbers is divisible by 4 or 6:

Product is divisible by 4 or 6 samples:

4 = (1 , 4) , ( 2, 2) , ( 4, 1)

6 = ( 1, 6) , ( 2, 3) , ( 3, 2) , ( 6 , 1)

8 = ( 1 , 8) , ( 2, 4) , ( 4 , 2) , ( 8 , 1)

12 = (2 , 6) , ( 3 , 4) , ( 4 3) , ( 6 , 2)

16 = (2 , 8) , ( 4 , 4) , ( 8 , 2)

18 = ( 3, 6) , ( 6 , 3)

20 = ( 4, 5) , ( 5, 4)

24 = ( 3 , 8) , ( 4 , 6 ) , ( 6 , 4) , ( 8 , 3)

28 = ( 4 , 7) , ( 7 , 4)

30 = ( 5 , 6) , ( 6 ,5 )

32 = ( 4 , 8) , ( 8 , 4)

36 = (6 , 6)

40 = ( 5 , 8) , ( 8 , 5)

42 = ( 6 , 7) , ( 7 , 6)

48 = ( 6 , 8) , ( 8 , 6)

56 = ( 7 , 8) , ( 8 , 8)

64 = ( 8 , 8)

Number of samples, n = 42

P (Product is less than or equal to 21) = 42/64 = 0.6563.

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Answers

Answer: see proof below

Step-by-step explanation:

Use the Power Reducing Identity: sin² Ф = (1 - cos 2Ф)/2

Use the Double Angle Identity: sin 2Ф = 2 sin Ф · cos Ф

Use the following Sum to Product Identities:

$\sin x - \sin y = 2\cos \bigg((x+y)/(2)\bigg)\sin \bigg((x-y)/(2)\bigg)\n\n\n\cos x - \cos y = -2\sin \bigg((x+y)/(2)\bigg)\sin \bigg((x-y)/(2)\bigg)$

Proof LHS → RHS

$\text{LHS:}\qquad \qquad \qquad (\sin^2A-\sin^2B)/(\sin A\cos A-\sin B \cos B)$

$\text{Power Reducing:}\qquad (\bigg((1-\cos 2A)/(2)\bigg)-\bigg((1-\cos 2B)/(2)\bigg))/(\sin A \cos A-\sin B\cos B)$

$\text{Half-Angle:}\qquad \qquad (\bigg((1-\cos 2A)/(2)\bigg)-\bigg((1-\cos 2B)/(2)\bigg))/((1)/(2)\bigg(\sin 2A-\sin 2B\bigg))$

$\text{Simplify:}\qquad \qquad (1-\cos 2A-1+\cos 2B)/(\sin 2A-\sin 2B)\n\n\n.\qquad \qquad \qquad =(-\cos 2A+\cos 2B)/(\sin 2A - \sin 2B)\n\n\n.\qquad \qquad \qquad =(\cos 2B-\cos 2A)/(\sin 2A-\sin 2B)$

$\text{Sum to Product:}\qquad \qquad (-2\sin \bigg((2B+2A)/(2)\bigg)\sin \bigg((2B-2A)/(2)\bigg))/(2\cos \bigg((2A+2B)/(2)\bigg)\sin \bigg((2A-2B)/(2)\bigg))$

$\text{Simplify:}\qquad \qquad (-2\sin (A + B)\cdot \sin (-[A - B]))/(2\cos (A + B) \cdot \sin (A - B))$

$\text{Co-function:}\qquad \qquad (2\sin (A + B)\cdot \sin (A - B))/(2\cos (A + B) \cdot \sin (A - B))$

$\text{Simplify:}\qquad \qquad \quad (\cos (A+B))/(\sin (A+B))\n\n\n.\qquad \qquad \qquad \quad =\tan (A+B)$

LHS = RHS: tan (A + B) = tan (A + B) $\checkmark$

Answer:

We know that,

$\dag\bf\:sin^2A=(1-cos2A)/(2)$

$\dag\bf\:sin2A=2sinA\:cosA$

___________________________________

Now, Let's solve !

$\leadsto\:\bf(sin^2A-sin^2B)/(sinA\:cosA-sinB\:cosB)$

$\leadsto\:\sf((1-cos2A)/(2)-(1-cos2B)/(2))/((2sinA\:cosA)/(2)-(2sinB\:cosB)/(2))$

$\leadsto\:\sf(1-cos2A-1+cos2B)/(sin2A-sin2B)$

$\leadsto\:\sf(2sin(2A+2B)/(2)\:sin(2A-2B)/(2))/(2sin(2A-2B)/(2)\:cos(2A+2B)/(2))$

$\leadsto\:\sf(sin(A+B))/(cos(A+B))$

$\leadsto\:\bf{tan(A+B)}$

What is the sum of r + s when r = –17.6 and s = 12.2? A. –29.8 B. –5.4 C. 5.4 D. 29.8

Answers

c. -5.4 -17.6+12.2= -5.4 because -17.6 goes first in the calculator but also its a negative

Richard's annual take-home pay is 28500. What is the maximum amount that he can spend per month paying off credit cards and loans and not be in danger of credit overload?

Answers

To answer the problem above, annual means yearly, to know the maximum amount that he can spend per month we simply divide the annual take-home pay with 12 months exactly. So 28500 / 12 = 2375. Richard can spend not more than 2375 to not be in danger of credit overload.

A mother decides to teach her son about a letter each day of the week. She will choose a letter from the name of the day. For example, on Saturday she might teach about the letter S or the letter U, but not the letter M. What letters are possible to teach using this method? (There are 15.) List all the letters in alphabetical order, with no spaces

Answers

Answer:

A, D, E, F, H, I, M, N, O, R, S, T, U, W, Y

Explanation:

Order the letters of each day, omitting the repetitions:

MONDAY:

A, D, M, N, O, Y

TUESDAY:

A, D, E, S, T, U, Y

WEDNESDAY:

A, D, E, N, S, W, Y

THURSDAY:

A, D, H, R, S, T, U, Y

FRIDAY:

A, D, F, I, R, Y

SATURDAY:

A, D, R, S, T, U, Y

SUNDAY:

A, D, N, S, U, Y

Put the letters together:

A, D, M, N, O, Y, A, D, E, S, T, U, Y, A, D, E, N, S, W, Y, A, D, H, R, S, T, U, Y, A, D, F, I, R, Y, A, D, R, S, T, U, Y, A, D, N, S, U, Y

Delete the repetitions:

A, D, M, N, O, Y, E, S, T, U, W, H, R, F, I.

Reorder:

A, D, E, F, H, I, M, N, O, R, S, T, U, W, Y ← answer

Determine the value of the expression 4^3 × 4^-6 .A) -12
B)-1/12
C)64
D)1/64

Answers

$\bf 4^3-4^(-6)\implies 4^(3-6)\implies 4^(-3)\n\n-----------------------------\n\nnow\quad recall\implies a^{-{ n}} \implies \cfrac{1}{a^( n)}\qquad \qquad\cfrac{1}{a^( n)}\implies a^{-{ n}}\n\n-----------------------------\n\n4^(-3)\implies \cfrac{1}{4^3}\implies \cfrac{1}{64}$