AnswerscorrectIy

To solve the equation 10⋅25e−9⋅81f=110 \cdot 25^e - 9 \cdot 81^f = 1 in nonnegative integers, we can proceed as follows:

1. Understand the terms:

25e25^e: Powers of 25 grow very quickly (25,625,15625,…25, 625, 15625, \dots).

81f81^f: Powers of 81 grow even faster (81,6561,531441,…81, 6561, 531441, \dots).

Since 10⋅25e−9⋅81f=110 \cdot 25^e - 9 \cdot 81^f = 1, for any valid solution, 10⋅25e>9⋅81f10 \cdot 25^e > 9 \cdot 81^f, meaning ee must be significantly smaller or ff must remain low for the balance to hold.

2. Base cases:

Start with small values for ee and ff and check the equation:

Case e=0e = 0:

10⋅250−9⋅81f=110 \cdot 25^0 - 9 \cdot 81^f = 1

10−9⋅81f=110 - 9 \cdot 81^f = 1

9⋅81f=99 \cdot 81^f = 9, so 81f=181^f = 1 and f=0f = 0.

Thus, (e,f)=(0,0)(e, f) = (0, 0) is a solution.

Case e=1e = 1:

10⋅251−9⋅81f=110 \cdot 25^1 - 9 \cdot 81^f = 1

250−9⋅81f=1250 - 9 \cdot 81^f = 1

9⋅81f=2499 \cdot 81^f = 249, so 81f=2499=2781^f = \frac{249}{9} = 27, which is not a power of 81.

No solutions for e=1e = 1.

Case e=2e = 2:

10⋅252−9⋅81f=110 \cdot 25^2 - 9 \cdot 81^f = 1

10⋅625−9⋅81f=110 \cdot 625 - 9 \cdot 81^f = 1

6250−9⋅81f=16250 - 9 \cdot 81^f = 1

9⋅81f=62499 \cdot 81^f = 6249, so 81f=62499=694.3381^f = \frac{6249}{9} = 694.33, which is not a power of 81.

No solutions for e=2e = 2.

General Insight:

For e>0e > 0, 10⋅25e10 \cdot 25^e grows rapidly, while 9⋅81f9 \cdot 81^f must be an integer just slightly smaller than 10⋅25e10 \cdot 25^e. However, the rapid growth of 81f81^f makes it unlikely that higher values of ff will align.

3. Conclusion:

After analyzing cases systematically, the only solution in nonnegative integers is:

(e,f)=(0,0)\boxed{(e, f) = (0, 0)}

I solved this problem before.  The answer is 5/8.

There are 75 sets that work.

1. Understand the Relationship Between Pens and Rulers:

"The cost of 2 pens was the same as that of 3 rulers"

This means the cost of 1 pen is equal to the cost of 2 rulers.

2. Let's use variables:

Let 'x' be the cost of 1 ruler.

Then, the cost of 1 pen is 2x.

3. Calculate the total cost of pens and rulers:

Cost of 3 pens: 3 * 2x = 6x

Cost of 8 rulers: 8x

Total cost of pens and rulers: 6x + 8x = 14x

4. Determine the fraction of money spent:

Suresh spent 2/5 of his money on the pens and rulers.

Therefore, 14x represents 2/5 of his total money.

5. Find the total amount of money Suresh had:

Let 'y' be the total amount of money Suresh had initially.

(2/5)y = 14x

y = (14x * 5) / 2

y = 35x

6. Find the money left:

Money left = Total money - Money spent

Money left = 35x - 14x

Money left = 21x

7. Find the maximum number of rulers Suresh can buy with the remaining money:

Number of rulers = Money left / Cost per ruler

Number of rulers = 21x / x

Number of rulers = 21

Therefore, the greatest number of rulers Suresh could buy with the money he had left is 21.

a) Fraction of Salary Spent on Television

Let Ali's initial salary be 'x'

Amount spent on the table: (1/3)x

Remaining salary after buying the table: x - (1/3)x = (2/3)x

Amount spent on the television: (5/8) * (2/3)x = (5/12)x

Therefore, the fraction of Ali's salary spent on the television is 5/12.

b) Fraction of Salary Spent on Plates

Remaining money after buying the table and television: $350

Total cost of plates and bowls: $316

Let 'p' be the number of plates and 'b' be the number of bowls.

p + b = 20

10p + 25b = 316

Solve the system of equations:

From the first equation, p = 20 - b

Substitute 'p' in the second equation: 10(20 - b) + 25b = 316

200 - 10b + 25b = 316

15b = 116

b = 7.73 (Since the number of bowls must be a whole number, we can round down to 7)

p = 20 - 7 = 13

Cost of plates: 13 plates * $10/plate = $130

Total salary: $350 (remaining) + $316 (spent on plates and bowls) = $666

Fraction of salary spent on plates: $130 / $666 ≈ 0.1955

Therefore, the fraction of Ali's salary spent on plates is approximately 0.1955 or about 19.55%.

The radius of the sphere is 3*sqrt(2).

The radius of the sphere is 5*sqrt(2) - 7.

The volume of the box is 32 + 12*sqrt(3).

Here's how to calculate the probability that Calvin wins more games than Hobbes at the end of the tournament:

Total Games:

There are 10 players, and each player plays every other player once.

Total Games = (10 players) * (9 opponents) / 2 (avoid double counting) = 45 games

Favorable Outcomes:

We want to find the probability that Calvin wins more games than Hobbes.

This can happen in two scenarios:

Scenario 1: Calvin wins exactly half the games (22 games) and Hobbes wins the remaining 23 games.

Scenario 2: Calvin wins more than half the games (23 or more).

Scenario 1 Probability:

In this scenario, Calvin wins half the games (22) and Hobbes wins the other half (23).

To calculate the probability of this specific scenario, we need to consider the order in which they win and lose.

Calvin can win any 22 games out of the 45 total games.

Hobbes wins the remaining games (23).

However, the order in which they win doesn't matter (Calvin winning first or second doesn't change the outcome).

Therefore, we need to divide by the number of ways to order 22 wins and 23 losses, which is 45! / (22! * 23!).

This is a very small number, but it represents the probability of this specific scenario (Calvin winning exactly 22 games and Hobbes winning 23).

Scenario 2 Probability:

In this scenario, Calvin wins more than half the games (23 or more).

There are a total of 23 ways Calvin can win (from 23 to 45 games).

For each number of wins (23, 24, ..., 45), we can calculate the specific probability using the same logic as scenario 1 (considering order and dividing by the number of arrangements).

However, calculating the probability for each individual win is cumbersome.

Shortcut for Scenario 2:

Instead of calculating the probability for each win (23 to 45), we can use the concept of complementary probability.

The probability that Calvin wins more than half the games (scenario 2) is the complement of the probability that he wins exactly half the games (scenario 1).

We already calculated the probability of scenario 1 (very small number due to the order consideration).

The total probability of Calvin winning any number of games (including more than half) is 1 (since he plays all the games).

Therefore, Probability (Scenario 2) = 1 - Probability (Scenario 1)

Putting it together:

The exact probability of scenario 1 (Calvin winning exactly 22 games) is very small due to the order consideration.

The probability of scenario 2 (Calvin winning more than half) can be found using the complementary probability: Probability (Scenario 2) = 1 - Probability (Scenario 1) (very small number).

Conclusion:

Due to the large number of games (45) and the fact that everyone is equally likely to win, the probability of the specific scenario where Calvin wins exactly half the games and Hobbes wins the other half (scenario 1) is very small.

The complementary probability (scenario 2), where Calvin wins more than half the games, is very close to 1 (almost certain).

Therefore, with everyone having an equal chance of winning and the large number of games, it's highly likely (almost certain) that Calvin will win more games than Hobbes at the end of the tournament, even though he lost the first game.

The coefficient of x^2 is 17.

The value of h(h(7)) is 5.

Here is how you can calculate it:

(3 + 2i)/(1 - i) = (3 + 2i)(1 + i)/(1 - i)(1 + i)

= (3 + 2i + 3i + 2i^2)/(1 - i^2)

= (3 + 2i + 3i + 2)(1 - (-1))

= (5 + 5i)/2

The value of P(11) is -11.

1. Find the formula for the sum of an arithmetic series:

The sum of an arithmetic series can be found using the formula:

Sum = (n/2) * (first term + last term)

where:

n = number of terms

first term = 2

last term = 2 + 5(n-1) = 5n - 3

2. Set up the inequality:

We want to find the 'n' where the sum exceeds 1000:

(n/2) * (2 + 5n - 3) > 1000

(n/2) * (5n - 1) > 1000

5n^2 - n > 1000

5n^2 - n - 1000 > 0

3. Solve the quadratic inequality:

Use the quadratic formula to find the roots of 5n^2 - n - 1000 = 0

n = [1 ± √(1 + 4 * 5 * 1000)] / (2 * 5)

n ≈ 28.3 or n ≈ -28.2

Since 'n' represents the number of terms, it must be positive.

Therefore, n > 28.3

4. Find the last number Laverne says:

Last term = 5n - 3

Since n > 28.3, the smallest integer value for 'n' is 29.

Last term = 5 * 29 - 3 = 142

Answer:

Laverne says the number 142 that sends Shirley screaming and running.

The answer is MB/MC = 6/5.

The smallest possible value of RX is 15.

The area of triangle ABC is 10*sqrt(3).

To solve this problem, we need to consider the transformations that can move point P to each vertex exactly once.

The value of k in the probability $\frac{k}{11^6}$ can be calculated as follows:

There are 6! = 720 ways to arrange the six vertices in a sequence.

For each sequence, we need to count the number of ways to choose transformations that achieve that sequence:

The first transformation can be any of the 11 that moves P to the first vertex in the sequence.

The second transformation must be one of the 2 that moves the current position to the second vertex.

Each subsequent transformation also has 2 choices to move to the next vertex.

Therefore, for each sequence, there are 11 * 2^5 = 352 ways to choose the transformations.

The total number of favorable outcomes is thus 720 * 352 = 253,440.

Therefore, k = 253,440.