Answer:
Step-by-step explanation:
(a) We will prove by induction that if n is a multiple of 3 then Bob has a winning strategy.
Let n=3
It is given that Alice always plays first.
Then, for the first move, Alice has a choice of removing 1 or 2 stones.
Case 1:
Alice removes 1 stone. Then it is now Bob's turn. There are 2 stones left. Bob has the choice of removing 1 or 2 stones. Then Bob's winning strategy should be to remove 2 stones. Then Bob removes the last stone and wins.
Case 2:
Alice removes 2 stones. Then it is Bob's turn. There is exactly 1 stone left, which Bob removes in his turn. Since he removes the last stone, he wins.
Thus, in either case, for n=3, Bob has a winning strategy. ____ (A)
Now, let us assume that Bob has a winning strategy for n=3p.
We will now check if Bob has a winning strategy for n=3p+3
It is given that Alice plays first.
Case 1:
Alice starts the game by removing 1 stone. Then, Bob has a choice of removing 1 or 2 stones. If he chooses to remove 2 stones, then the number of stones left is 3p+3-3=3p and it is now Alice's turn to play. This is exactly the game when n=3p and we have already assumed that Bob has a winning strategy for n=3p. Thus, in this case Bob has a winning strategy.
Case 2:
Alice starts the game by removing 2 stones. Then, Bob has a choice of removing 1 or 2 stones. If he chooses to remove 1 stone, then the number of stones left is 3p+3-3=3p and it is now Alice's turn to play. This is again exactly the game when n=3p and we have already assumed that Bob has a winning strategy for n=3p. Thus, in this case also, Bob has a winning strategy.
Thus, in either case Bob has a winning strategy for n=3p+3 if he has a winning strategy for n=3p. _______ (B)
Thus, from (A) & (B), using induction, we can say that Bob has a winning strategy if n is a multiple of 3.
(b) We will now prove that if n is not a multiple of 3, then Alice has a winning strategy.
If n is not a multiple of 3, then n can have either of the forms of 3m+1 or 3m+2, m ∈ W.
We will prove the given fact for both the forms simultaneously
Let m=0, i.e., n=1 or n=2
Since Alice starts first, she removes 1 stone, if n=1 or 2 stones if n=2 and thus wins. Thus Alice has a winning strategy if m=0. ______ (A)
Let us assume that Alice has a winning strategy for m=k, i.e., for n=3k+1 & n=3k+2
Now, we will check if Alice has a winning strategy for m=k+1, i.e., for n=3(k+1)+1=3k+4 or n=3(k+1)+2=3k+5
Let n=3k+4
Since Alice plays first, she has a choice to remove 1 or 2 stones.
Note that, if Alice removes 2 stones and in turn Bob removes 2 stones, then the number of stones becomes a multiple of 3 such that it is Alice's turn to play. In that case, Bob will have a winning strategy as shown in the previous part.
Then, Alice should remove 1 stone. Then, it is now Bob's turn and he has a choice of removing 1 or 2 stones.
Case 1:
Bob removes 1 stone. Then there are 3k+4-1-1=3k+2 stones remaining and it is Alice's turn. This is identical to the game where n=3k+2. We have already assumed that Alice has a winning strategy in this case.
Case 2:
Bob removes 2 stones. Then there are 3k+4-1-2=3k+1 stones remaining and it is Alice's turn. This is identical to the game where n=3k+1. We have already assumed that Alice has a winning strategy in this case.
Thus, in either case, Alice has a winning strategy.
Let n=3k+5
Since Alice plays first, she has a choice to remove 1 or 2 stones.
Note that, if Alice removes 1 stone and in turn Bob removes 1 stone, then the number of stones becomes a multiple of 3 such that it is Alice's turn to play. In that case, Bob will have a winning strategy as shown in the previous part.
Then, Alice should remove 2 stones. Then, it is now Bob's turn and he has a choice of removing 1 or 2 stones.
Case 1:
Bob removes 1 stone. Then there are 3k+5-2-1=3k+2 stones remaining and it is Alice's turn. This is identical to the game where n=3k+2. We have already assumed that Alice has a winning strategy in this case.
Case 2:
Bob removes 2 stones. Then there are 3k+5-2-2=3k+1 stones remaining and it is Alice's turn. This is identical to the game where n=3k+1. We have already assumed that Alice has a winning strategy in this case.
Thus, in either case, Alice has a winning strategy.
Then, we can say that Alice has a winning strategy for m=k+1 if she has a winning strategy for m=k. _____ (B)
Then, by induction, from (A) & (B), we can say that if n is not a multiple of 3, then Alice has a winning strategy.
I GIVE BRAINLIEST!!! Derek wants to save up for a new baseball glove. The price of the glove with tax is $385. If he can save $65 per month , how many months will it take him to save up for the glove?
Answer:
6
Step-by-step explanation:
385/65= about 6 but technically its like 5.92307...
Step-by-step explanation:
Price of glove with tax = 385
Money saved per month = 65
Months needed for Derek to save enough money
= 385/65
= 5.923 months
Solve for x
Solve for x give that the segments ST and SU are both
tangent to the circle.
T
7X-3
Submit
5x+1
U
Answer:
2
Step-by-step explanation:
ST and SU are tangents to the [tex] \odot L[/tex] from the point S to the points T and U respectively. Measures of the tangents drawn from external point to a circle are equal. [tex] \therefore ST = SU[/tex] [tex] \therefore 7x - 3= 5x+1[/tex] [tex] \therefore 7x - 5x=3+1[/tex] [tex] \therefore 2x=4[/tex] [tex] \therefore x=\frac{4}{2}[/tex] [tex] \therefore x={2}[/tex]Please someone help me
The U.S. Energy Information Administration (US EIA) reported that the average price for a gallon of regular gasoline is $2.94. The US EIA updates its estimates of average gas prices on a weekly basis. Assume the standard deviation is $0.20 for the price of a gallon of regular gasoline and recommend the appropriate sample size for the US EIA to use if they wish to report each of the following margins of error at 95% confidence. (Round your answers up to the nearest whole number.) (a) The desired margin of error is $0.10. 16 Changed: Your submitted answer was incorrect. Your current answer has not been submitted. (b) The desired margin of error is $0.06. 43 Correct: Your answer is correct. (c) The desired margin of error is $0.04.
Answer:
The answer is "16, 43, and 96".
Step-by-step explanation:
Given:
[tex]\sigma = 0.20\\\\c = 95\% = 0.95\\\\\therefore \alpha = 1- c = 1- 0.95 = 0.05\\\\\therefore \frac{\alpha}{2} = 0.025\\\\[/tex]
Using Z table:
[tex]\therefore Z_{\frac{\alpha}{2}} = 1.96\\\\[/tex]
For point a:
[tex]E = 0.10\\\\n=(\frac{Z_{\frac{\alpha}{2} \times \sigma}}{E})^2[/tex]
[tex]= (\frac{(1.96\times 0.20)}{0.10})^2\\\\= 15.3664 \approx 16[/tex]
so, The Sample size (n) = 16
For point b:
[tex]E = 0.06\\\\n=(\frac{Z_{\frac{\alpha}{2} \times \sigma}}{E})^2[/tex]
[tex]= (\frac{(1.96\times 0.20)}{0.06})^2\\\\= 42.6844444444\approx 43[/tex]
For point c:
[tex]E = 0.04\\\\n=(\frac{Z_{\frac{\alpha}{2} \times \sigma}}{E})^2[/tex]
[tex]= (\frac{(1.96\times 0.20)}{0.04})^2\\\\= 96.04\approx 96[/tex]
PLS ANWER ASAP!! person with correct answer will be marked brainliest
Answer:
62 degrees since there are supplementary angles
Step-by-step explanation:
∠PQO= 180 degrees - 118 degrees
∠PQO= 62 degrees is the answer.
∠PQO= 180°-∠RQO -------(Supplementary Angles)
∠PQO= 180° - 118°
∠PQO= 62° answer.
Need help will give brainiest
Answer:
3. x=50, y=80, 4. x=6.5, y=5
Step-by-step explanation:
I think, not entirely sure.
I don’t think this is correct, accidentally clicked. Thanks in advance.
A laptop computer is purchased for $3900. Each year, its value is 75% of its value the year before. After how many years will the laptop computer be worth $700 or less?
Answer:
1200*0.70=840
840*0.70=588
588*0.70=411.60
411.60*0.70=288.12
PLS ITS URGENT ILL GIVE 70 POINTS AND BRAINLIEST
ANY TROLL ANSWERS WILL GET REPORTED
Mr. and Mrs. Pinkerton have four boys. Every time people notice this they comment about the odds of this happening. The Pinkertons decide to set up a simulation with 10 trials to
determine the probability.
Answer these three questions with 1-2 sentences
a. What simulation could be used to represent the likelihood of having a boy or a girl?
b. Which outcome represents having a boy? Which outcome represents having a girl?
c. How should the simulation be designed in order to account for four different children?
Answer:
a. Flipping a coin
b. heads ⇒ boy tails ⇒ girl
c. Flip 4 coins (1 for each child) ten times
Step-by-step explanation:
Venny Had 5 friends she ate 3 of them. The rest ran away. How much friends dose she have now?
PLS HELP!! need the answer asap
Evaluate the following expression. 45%[6x(17-9)-30]
Answer:
tdxutdxfhdxfhx hd tudbtd tudbytd
Step-by-step explanation:
fc fcgbtufnyufnutfiygniygmyufniygnyigfntudb rsdut rd utddutestur dutrd uttu uhgugfutfuy8. f utfuy. dutugvu v gigv. uvv gihg ihg igg gv ugv vug ugf ugv ugv v igiugigougojgohgiohgk h. i gigiy gygiygiytiy
Your friend says she has an unfair number cube: the probability of getting a one or a six is 1/3 for each, and the probability of getting a two, three, four, or five is 1/12 for each. You want to test her statement. If you roll the number cube 96 times, what are the expected values for each number on the number cube
Answer:
See below.
Step-by-step explanation:
Expected number = probability x 96 (the number of rolls)
Outcome Probability Expected Value
1 1/3 (1/3)(96) = 32
2 1/12 8
3 1/12 8
4 1/12 8
5 1/12 8
6 1/3 32
Find the volume of cone pictured below. Use 3.14 for /pi Round your answer to the nearest hundredth.
Answer:
134.04
hope this helps
have a good day :)
Step-by-step explanation:
Which graph represents a proportional relationship? On a coordinate plane, a straight line with positive slope goes through points (0, 0), (1, 2), and (2, 4). On a coordinate plane, a vertical line is at x = negative 3. On a coordinate plane, a horizontal line is at y = 3. On a coordinate plane, a curved line goes through the origin.
Answer:
Step-by-step explanation:
D. On a coordinate plane, a straight line with positive slope goes through points (3, 3) and (4, 4).
Explanation:
For a relationship to be proportional it needs to meet two conditions:
1) its graph needs to be a straight line
2) every point of the graph needs to satisty the equation y = k•x, or, in other words k has to be equal for every point
Now, let's check the answers;
A) we have two points: A (2, 6) and B (4, 5). For point A, x is 2 and y is 6, so:
y = k•x
6 = k•2
k = 3
For point B, x is 4 and y is 5, so:
y = k•x
5 = k•4
k= 5/4
Since k isn't the same value for these two points, this isn't proprtional relationship.
B) this graph is curve, so since it's not a straight line it can't be proportional relationship.
C) similarly to graph A we have point A (5, 2) and point B (6, 3).
For point A, x is 5 and y is 2, so:
y = k•x
2 = k•5
k = 2/5
For point B, x is 6 and y is 3, so:
y = k•x
3 = k•6
k = 1/2
Again values of k aren't the same, so this isn't a proportional relationship.
D) Point A (3, 3) and point B (4, 4). For point A, x is 3 and y is 3, so:
y = k•x
3 = k•3
k = 1
For point B, x is 4 and y is 4, so:
y = k•x
4 = k•4
k = 1
Since value of k is the same for both points, this graph shows the proportional relationship.
Answer:
a
Step-by-step explanation: the line goes thru the origin
What is the volume, in cubic inches, of a cylinder with a height of 10 inches and a base radius of 2 inches, to the nearest tenths place?
Answer:
125.7 inches cubed.
Step-by-step explanation:
πr^h is the formula for volume of cylinder.
The question is in the picture (Alegbra 1)
Answer:
[tex] \frac{s}{s + 3} [/tex]
Answer:
S/S+3
Step-by-step explanation:
Hope this helps...
(10+4×4×4×4+22)÷4/8?? what is the answer??
Answer:
5164
Step-by-step explanation:
Answer: 576
Step-by-step explanation:
4x4x4x4 = 256
10+ 256 +22
288 divided by 4/8
576
Which inequality’s have the same solution set graphed on the number line? Select two options.
Answer:
x < -4-4 > xExplanation:
Both of these inequalities are describing that the value of x is less than negative four, as observable by the solution set graphed on the number line.
Type SSS, SAS, ASA, AAS, or HL to justify why the two larger triangles are congruent. B с AC = DB D Enter
Answer:
HL
Step-by-step explanation:
The two larger triangles shown in the diagram above each has a corresponding leg that is congruent to each other (BC = BC).
Also, we are given that the hypotenuse AC of one is congruent to the hypotenuse DB of the other.
This satisfy the Hypotenuse-Leg Congruence Theorem. Thus, the two larger triangles are congruent based on the HL Congruence Criterion which says two triangles are congruent if the leg and hypotenuse of one is congruent to the corresponding leg and hypotenuse of the other.
How many 1/2 unit cubes does it take to fill the prism?
lcm of 64 and 94
plz show photo of work and explan plz for 10
Answer:
10 and 64 and is a then not answer
Answer:
Step-by-step explanation:
2 64, 94
32, 47
LCM=2*32*47=3008
while doing LCM u should take all the numbers (common numbers and remaining numbers)
But in HCF only common numbers are take for eg .here,HCf is 2because 2 is the only number that divides both the number.
PLEASE HELP!!
with
5,6,7
z’s options are 4,22,8,11
i really need help i’ve been stuck in this for way to long
=======================================================
Explanations:
Problem 5)
Assuming HEFG is a parallelogram, this means the opposite sides are the same length.
EH = GF
x-3 = 4x-15
x-4x = -15+3
-3x = -12
x = -12/(-3)
x = 4
--------------------------
Problem 6)
We'll use the same idea from problem 5. The opposite sides EF and HG are congruent
EF = HG
3y = 3x+12
3y = 3*4+12
3y = 24
y = 24/3
y = 8
--------------------------
Problem 7)
For any parallelogram, the diagonals always cut each other in half.
Therefore HK = 2z+3 is exactly half that of segment HF = 22
Put another way, HF is twice as long as HK
2*(HK) = HF
2*(2z+3) = 22
4z+6 = 22
4z = 22-6
4z = 16
z = 16/4
z = 4
Which statements describe one of the transformations performed on f(x) = x2
to create g(x)= 3(x+ 5)2 -2 ? Choose all that apply.
Given:
The given functions are:
[tex]f(x)=x^2[/tex]
[tex]g(x)=3(x+5)^2-2[/tex]
To find:
The transformations performed of f(x) to create g(x).
Solution:
The translation is defined as
[tex]g(x)=kf(x+a)+b[/tex] .... (i)
Where, k is stretch factor, a is horizontal shift and b is vertical shift.
If 0<k<1, then the graph compressed vertically by factor k and if k>1, then the graph stretch vertically by factor k.
If a>0, then the graph shifts a units left and if a<0, then the graph shifts a units right.
If b>0, then the graph shifts b units up and if b<0, then the graph shifts b units down.
We have,
[tex]f(x)=x^2[/tex]
[tex]g(x)=3(x+5)^2-2[/tex]
Using these two function, we get
[tex]g(x)=3f(x+5)-2[/tex] ...(ii)
On comparing (i) and (ii), we get
[tex]k=3,a=5,b=-2[/tex]
It means the graph of f(x) is vertically stretched with a scale factor of 3, shifts 5 units left and 2 units down to get g(x).
Therefore, the correct options are A and C.
Looking in her food pantry, Professor Martinez finds a package of 12 flavored water bottles. The package contains 5 strawberry flavored water bottles and 7 mango flavored water bottles, each without labeling. All the bottles look alike and have equal probability of being chosen. Three bottles are randomly selected without replacing them.
a. The probability that all (exactly three) bottles are strawberry flavored is: _______
b. The probability that at least one of the three bottles are strawberry flavored is: _________
c. The probability that exactly 2 bottles are strawberry flavored is: _________
Answer:
a) 12/24 * 11/24*10/24
b) 1 - 12/24 * 11/24*10/24
c) 12/24 * 11/24 *12/24
Step-by-step explanation:
Total number of bottles = 12 + 5+7 = 24
a. The probability that all (exactly three) bottles are strawberry flavored is: 212
12/24 * 11/24*10/24
b. The probability that at least one of the three bottles are strawberry flavored is:
1 - P (no strawberry bottle)
1 - (12/24*11/24*10/24)
c. The probability that exactly 2
12/24 * 11/24 *12/24
Write the equation of the circle centered at ( - 4,7) with radius 20?
Answer:
(x + 4)² + (y - 7)² = 20²
Step-by-step explanation:
Graphing form
(x - h)² + (y - k)² = r²
(h, k) is the center = (-4, 7)
r is the radius = 20
-----------------------------------
Plug in the givens
(x + 4)² + (y - 7)² = 20²
A puppy weighs534pounds.
How many ounces does the puppy weigh?
Answer:
I believe that the question is "A puppy weighs 5 3/4 pounds. How many ounces does the puppy weigh."
Step-by-step explanation:
The answer is 23 ounces.
Find the value of h.
Question 6 options:
A)
51.995
B)
28.321
C)
62.003
D)
95.476
Answer:
D. 95.476.
Step-by-step explanation:
sin 33 = 52/h
h = 52/sin 33
h = 95.476.
Answer:
Step-by-step explanation:
We will use right triangle trig and SohCahToa here to solve this. The side across from the angle marked as 33 degrees is given as 52; we are looking for h, the hypotenuse. The trig ratio that uses the the side across from the reference angle along with the hypotenuse is the sin ratio. Setting up the equation:
[tex]sin33=\frac{52}{h}[/tex]. Multiply both sides by h and then divide by sin(33) to get
[tex]h=\frac{52}{sin33}[/tex] so
h = 95.476, choice D.
Make sure your calculator is in degree mode and not radians when your angles are given in degrees
I need answer to both of the questions please someone!! 20points
Answer:
Step-by-step explanation:
x+62=90 degree (being perpendicular)
x=90-62
x=28 degree
x+25=90 (being perpendicular)
x=90-25
x=65 degree
Answer:
1)X+62 =90
x+62-62=90-62
x =28
2) X+25+90+180 =360
x +295=360
x =360-295
x =65
The number of blue fish to red fish in a tank is
proportional. If there are 34 blue fish and 50
red fish in Tank A, how many blue fish are in
Tank B if there are 75 red fish?