What is algebra? Why is it so crucial for mathematics?

The probability that the mean of a sample of 77 computers would be greater than 82.59 months, assuming the population mean is 80 months and the variance is 64, is approximately 0.0606

The situation described can be modeled using a normal distribution, with a mean of 80 months and a standard deviation of the square root of the variance, which is 8 months (since variance = standard deviation squared).

To find the probability that the mean of a sample of 77 computers would be greater than 82.59 months, we need to standardize the sample mean using the formula

z = (x - μ) / (σ / √n)

where

x is the sample mean

μ is the population mean (believed to be 80 months)

σ is the population standard deviation (8 months)

n is the sample size (77)

Plugging in the values, we get

z = (82.59 - 80) / (8 / √77) ≈ 1.55

To find the probability of a z-score being greater than 1.55, we can use a standard normal distribution table or calculator. From the table, we find that the probability of z being greater than 1.55 is approximately 0.0606.

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To find the inverse of a function, we need to swap the positions of x and y and then solve for y. In other words, we replace f(x) with y and then solve for x.

x = 5y - 6

Next, we'll solve this equation for y:

x + 6 = 5y

y = (x + 6)/5

The decimal that represents the fraction of the $30.00 Maggie spent is 0.6.

Now, let's talk about decimals. Decimals are a way of expressing parts of a whole number in a fraction of 10. For example, 0.5 is the same as 1/2. In your situation, Maggie spent $18.00 out of $30.00. To figure out what decimal represents the fraction of the $30.00 Maggie spent, we need to divide the amount she spent by the total amount she had.

So, we can write this as a fraction:

$18.00 / $30.00

To turn this fraction into a decimal, we divide the numerator (top number) by the denominator (bottom number) using long division or a calculator.

$18.00 / $30.00 = 0.6

Another way to say this is that Maggie spent 60% of the money she had in her wallet.

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To reduce the width of a confidence interval, you can use a smaller sample size and decrease the level of confidence, or use a larger sample size and decrease the level of confidence.

What are the factors?

what are factorsIn mathematics, a factor is a number that divides another number without leaving a remainder.

The combination of factors that would definitely cause the confidence interval to become wider is to use a larger sample and increase the level of confidence or to use a smaller sample and increase the level of confidence.

When using a larger sample size, the standard error of the mean decreases, and the interval will become narrower. Conversely, when using a smaller sample size, the standard error of the mean increases, and the interval will become wider. However, increasing the level of confidence will also increase the width of the interval as a wider interval is required to capture the true population parameter with a higher level of confidence.

Therefore, to reduce the width of a confidence interval, you can use a smaller sample size and decrease the level of confidence, or use a larger sample size and decrease the level of confidence.

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The actual distance between the two towns is 337.5 miles.

What is distance?

Distance is the measure of how far apart two objects or locations are from each other. It is usually measured in units such as meters, kilometers, miles, or feet. Distance is a scalar quantity, meaning it has only magnitude and no direction.

If 1/2 inch on the map represents 75 miles in reality, then 1 inch on the map would represent 150 miles (twice the distance of 1/2 inch).

So, to find the actual distance between the two towns, we need to multiply the distance on the map (2 1/4 inches) by the conversion factor (150 miles per inch):

2 1/4 inches * 150 miles per inch = (9/4) inches * 150 miles per inch

Simplifying:

(9/4) inches * 150 miles per inch = 337.5 miles

Therefore, the actual distance between the two towns is 337.5 miles.

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Answer:

Same equation just using the assocaitive property

Step-by-step explanation:

For example, 8 + (2 + 3) = (8 + 2) + 3 = 13

Hope this helps! =D

The probability of someone randomly selecting a computer that does not have a virus is 0.8.

Total number of computers = 20

Virus affected = 4

Perfect computers = 20 - 4 = 16

The probability of choosing a computer that does not have a virus is equivalent to the number of computers without a virus and divided by the total number of computers in the lab.

The probability of selecting a computer without a virus is:

P(C) = 16/20

Simplifying this fraction, we get:

P(20 - 4 = 16) = 4/5

P(20 - 4 = 16) = 0.8

Therefore we can conclude that the probability of someone randomly selecting a computer that does not have a virus is 0.8.

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Saanvi paid £5329.39 for the boat 13 years ago. This is based on the assumption that the boat depreciated at a constant rate of 2.5% per year. It is important to note that there could be other factors that affect the value of the boat, such as maintenance, repairs, and upgrades, which are not accounted for in this calculation.

How to solve the question?

To find out how much Saanvi paid for the boat, we can use the concept of depreciation. Depreciation is the reduction in value of an asset over time. In this case, the boat has depreciated at a rate of 2.5% per year.

Let the initial value of the boat be 'P'. After one year, the value of the boat would be 97.5% of P. After two years, it would be 95% of P, and so on. We can write this mathematically as:

Value of the boat after n years = P x (0.975)^n

We are given that the value of the boat after 13 years is £3115. Therefore, we can write:

3115 = P x (0.975)^13

Solving this equation for P, we get:

P = 3115 / (0.975)^13

P = 5329.39

Therefore, Saanvi paid £5329.39 for the boat.

In conclusion, Saanvi paid £5329.39 for the boat 13 years ago. This is based on the assumption that the boat depreciated at a constant rate of 2.5% per year. It is important to note that there could be other factors that affect the value of the boat, such as maintenance, repairs, and upgrades, which are not accounted for in this calculation.  

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The value of m∠1 in the given diagram is 113 (last option)

From the diagram given above, we obtained the following data:

Considering the above, we can easily obtain the value of m∠1 as follow:

m∠1 + 67 = 180 (angle on a straight line)

Collect like terms

m∠1 = 180 - 67

m∠1 = 113

Thus, we can conclude from the above calculation that the value of m∠1 is 113 (last option)

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The sales tax rate is 6%

If the sales-tax rate is 6%, the tax paid on an $8.50 purchase would be:

Tax = 0.06 x $8.50

Tax = $0.51

Therefore, the total cost of the purchase including sales tax would be:

Total cost = $8.50 + $0.51

Total cost = $9.01

The sales tax rate is 6%.

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Answer:

Vertex = (-3, -4)

Step-by-step explanation:

The given graph is a parabola that opens upwards.

The vertex of a parabola that opens upwards is its lowest point (minimum value).

From inspection of the given graph, the lowest point is (-3, -4).

Therefore, the vertex of the parabola is (-3, -4).

By using Pythagorean theorem, we find that the shortest distance from Krista's home to her work is approximately 1.19 miles.

To find the shortest distance from Krista's home to her work, we can use the Pythagorean theorem. Let's convert the distances to a common unit, such as miles

Distance from grocery store to work = 500 m = 0.31 miles (since 1 mile = 1609.34 meters)

Total distance from home to work = 2 km = 1.24 miles (since 1 mile = 1.609 km)

Let's call the distance from Krista's home to the grocery store "x". Then, we can set up the following equation

x^2 + 0.31^2 = 1.24^2

Simplifying and solving for x, we get

x^2 = 1.24^2 - 0.31^2 = 1.4223

x = √1.4223 ≈ 1.19 miles

Therefore, the shortest distance is approximately 1.19 miles.

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--The given question is incomplete, the complete question is given

" Krista needs to stop at the grocery store on the way home from her work. What is the shortest distance, in miles, from Krista's home to her work? the distance from grocery store to work is 500m, and the total distance from home to work is 2km"--

Step-by-step explanation:

'Range' is the 'y' values a graph can have....

 this one goes from a high of -8   down through - inf

-8 <= y < -inf

(-inf, -8]  

There are 73,701 different 6-card hands (from a standard deck) with at least 3 kings.

To calculate the number of 6-card hands with at least 3 K's, the problem can be divided into:

Case 1:

Exactly 3 Kings

There are 4 ways to choose 3 kings to put in the hand, then there are 48 cards left to choose the remaining 3 cards (because we used 3 cards in a 52-card deck). Therefore, the number of 6-card hands with exactly 3 kings is:

4 * (48 choose 3) = 4 * 17,296 = 69,184

Case 2:

Exactly 4 Kings

There are 4 ways to choose 4 kings to put in the hand, then there are 48 cards left to choose the remaining 2 cards. Therefore, the number of 6-card hands with exactly 4 kings is:

4 * (48 choose 2) = 4 * 1.128 = 4.512

Case 3:

Exactly 5 kings

There are 4 ways to choose the 5 kings in the hand, then there is only one card left to choose from (because we used 5 of the 52 cards in the deck of cards). Therefore, the number of 6-card hands with exactly 5 kings is:

4*1=4

Case 4:

6 cards are king

There is only one way to choose all 6 cards as king.

Therefore, the total number of 6-card hands with at least 3 kings is:

69,184 + 4,512 + 4 + 1 = 73.701

So there are 73,701 different 6-card hands with at least 3 kings.  

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Answer: -12d+(4/(4d+7))

Step-by-step explanation:

To simplify the expression –48d^2 – 132d – 80 ÷ (4d + 7), we can use long division:

           -12d

      ____________

   4d + 7 | -48d^2 - 132d - 80

            48d^2 + 84d

            ___________

                       - 48d - 80

                       + 48d + 84

                       ___________

                                4

Therefore, the simplified expression is -12d + (4/(4d + 7))

Answer:

x = 4√5 ≈ 8.94 (2 d.p.)

y = 8√5 ≈ 17.89 (2 d.p.)

Step-by-step explanation:

To find the values of x and y, use the Geometric Mean Theorem (Leg Rule).

The altitude drawn from the vertex of the right angle perpendicular to the hypotenuse separates the hypotenuse into two segments. The ratio of the hypotenuse to one leg is equal to the ratio of the same leg and the segment directly opposite the leg.

[tex]\boxed{\sf \dfrac{Hypotenuse}{Leg\:1}=\dfrac{Leg\:1}{Segment\;1}}\quad \sf and \quad \boxed{\sf \dfrac{Hypotenuse}{Leg\:2}=\dfrac{Leg\:2}{Segment\;2}}[/tex]

From inspection of the given right triangle RST:

Substitute the values into the formulas:

[tex]\boxed{\dfrac{20}{y}=\dfrac{y}{16}}\quad \sf and \quad \boxed{\dfrac{20}{x}=\dfrac{x}{4}}[/tex]

Solve the equation for x:

[tex]\implies \dfrac{20}{x}=\dfrac{x}{4}[/tex]

[tex]\implies 4x \cdot \dfrac{20}{x}=4x \cdot \dfrac{x}{4}[/tex]

[tex]\implies 80=x^2[/tex]

[tex]\implies \sqrt{x^2}=\sqrt{80}[/tex]

[tex]\implies x=\sqrt{80}[/tex]

[tex]\implies x=\sqrt{4^2\cdot 5}[/tex]

[tex]\implies x=\sqrt{4^2}\sqrt{5}[/tex]

[tex]\implies x=4\sqrt{5}[/tex]

Solve the equation for y:

[tex]\implies \dfrac{20}{y}=\dfrac{y}{16}[/tex]

[tex]\implies 16y \cdot \dfrac{20}{y}=16y \cdot \dfrac{y}{16}[/tex]

[tex]\implies 320=y^2[/tex]

[tex]\implies \sqrt{y^2}=\sqrt{320}[/tex]

[tex]\implies y=\sqrt{320}[/tex]

[tex]\implies y=\sqrt{8^2\cdot 5}[/tex]

[tex]\implies y=\sqrt{8^2}\sqrt{5}[/tex]

[tex]\implies y=8\sqrt{5}[/tex]

The probability that Alyse has a smaller time than Jocelyn in a 1 mile race on a randomly selected day is approximately 0.2676

To find the probability that Alyse has a smaller time than Jocelyn in a 1 mile race on a randomly selected day, we need to compare the distribution of their running times.

Let X be the running time of Alyse and Y be the running time of Jocelyn. Then, we have

X ~ N(13.5, 2.5^2)

Y ~ N(12, 1.5^2)

We want to find P(X < Y). We can start by standardizing the variables:

Zx = (X - 13.5) / 2.5

Zy = (Y - 12) / 1.5

Then, we have

P(X < Y) = P(X - Y < 0)

Substituting the standardized variables, we get

P(X - Y < 0) = P((Zx - Zy) < (0 - (13.5-12)/sqrt(2.5^2 + 1.5^2)))

Using the standard Normal distribution table or calculator, we find that the probability of Zx - Zy being less than -0.624 is approximately 0.2676.

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The next time the walking and bus tours will start at the same time is 11:45 a.m.

What is the lcm?

The LCM is multiple which is useful if fractions need to be expressed in the same name, when the other number is multiple, LCM will have the larger number:

To find out when the walking and bus tours will start at the same time, we need to find the least common multiple (LCM) of 25 and 45, which is the smallest time interval that is a multiple of both 25 minutes and 45 minutes.

The prime factorization of 25 is 5 * 5, and the prime factorization of 45 is 3 * 3 * 5. To find the LCM, we take the highest power of each prime factor that appears in either factorization, so:

LCM = 3 * 3 * 5 * 5 = 225

Therefore, the walking and bus tours will start at the same time every 225 minutes, or 3 hours and 45 minutes. To find the next time they will start at the same time, we need to add 225 minutes to the starting time of 8:00 a.m.

8:00 a.m. + 3 hours and 45 minutes = 11:45 a.m.

Hence, the next time the walking and bus tours will start at the same time is 11:45 a.m.

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The corrected statement should look like this:
```java
int[] hours = {8, 12, 16};
```

It looks like you are trying to create an integer array in Java called "hours" and initialize it with the values 8, 12, and 16. The given statement has incorrect syntax.
Declare the integer array using "int[]" followed by the variable name "hours".

This tells the Java compiler that you want to create an array of integers called "hours".
Use the assignment operator "=" to assign values to the array.
Enclose the values you want to assign to the array within curly braces "{" and "}" and separate each value with a comma.
The corrected statement should look like this:
```java
int[] hours = {8, 12, 16};
```
Now, the "hours" integer array is initialized correctly with the given values.

The syntax for declaring and initializing an integer array in Java is:

```java
data Type[] array

Name = {value1, value2, value3, ...};
```
In this case, "data Type" is "int", "array

Name" is "hours", and the values are 8, 12, and 16.
Remember to keep your code clean and concise to avoid syntax errors and make it easier for others to understand.

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(13) Angle A, angle B and angle C are collinear and are proved.

(14) Based on corresponding angles theorem, X is bisector of angle A.

The sum of the measures of collinear angles is always 180 degrees, as they together form a straight angle.

If we consider triangle PCQ;

Since line CP = line CQ; then angle P = angle Q = x

m∠PCQ = 180 - 2x

If we consider triangle PBQ;

Since line PB = line BQ; then angle P = angle Q = x

m∠PBQ = 180 - 2x

If we consider triangle PAQ;

Since line AP = line AQ; then angle P = angle Q = x

m∠PAQ = 180 - 2x

Thus, angle A, angle B and angle C are collinear.

14. To prove that X is a bisector A;

Draw a line parallel to AN from point B to opposite side, call this point Y

Angle formed from line M to line BY, is a corresponding angle to angle formed from line M to line AN.

If X is the bisector of angle B, then it is also a bisector angle A since both angles are equal.

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Answer:

The answer that you're looking for is approximately 9202.77 and in terms of π it is 2929.33π

Step-by-step explanation:

Using the equation [tex]\frac{4}{3}\pi r^{3}[/tex] you can replace r with 13 to get [tex]\frac{4}{3} \pi 13^{3}[/tex] you then multiply them all to get 9202.77 and divide by π to find the terms of pi which is 2929.33π.

I hope this was helpful!

Answer:

The answer is 3

Step-by-step explanation:

x×108=y

x×2²×3³=y

3×108=324

Arun is not correct - he has just under $5 in total, not just over.

Let's start by finding out how many 5-cent and 10-cent coins Arun has.

Let the number of 5-cent coins be 5x and the number of 10-cent coins be 3x (since the coins are in the ratio 5:3).

Then the total value of the 5-cent coins is 5x0.05 = 0.25x dollars, and the total value of the 10-cent coins is 3x0.1 = 0.3x dollars.

So the total value of all the coins is 0.25x + 0.3x = 0.55x dollars.

Since Arun has 72 coins, we know that 5x + 3x = 72, or 8x = 72, or x = 9.

Therefore, Arun has 5x = 59 = 45 5-cent coins and 3x = 39 = 27 10-cent coins.

The total value of these coins is 450.05 + 270.1 = 2.25 + 2.7 = 4.95 dollars.

So Arun is not correct - he has just under $5 in total, not just over.

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Answer:

mean: 101 (add all the numbers then divide by 7)

mode: 102 (the most frequent number in the set)

median: 102 (the number in the middle of the set)

range: 20 (the difference between the largest and smallest number)

Mean = 101

Mode = 102

Median = 102

Range = 20

MEAN: Add up all the numbers, then divide by how many numbers there are.

102 + 107 + 99 + 102 + 111 + 95 + 91 = 707

707 ÷ 7 = 101

MODE: Arrange all numbers in order from lowest to highest or highest to lowest and then count how many times each number appears in the set. The one that appears the most is the mode.

91,95,99,102,102,107,111

MEDIAN: Arrange the numbers from smallest to largest. If the amount of numbers is odd, the median is the middle number. If it is even, the median is the average of the two middle numbers in the list.

91,95,99,102,102,107,111

RANGE: Subtract the lowest number from the highest number

111 - 91 = 20

a) The probability that the third hole drilled is the first to yield a productive well is given by the following sequence of events: the first two holes must be unproductive, followed by a productive third hole. The probability of each of these events happening is:

- Probability of an unproductive hole: 0.8

- Probability of two unproductive holes in a row: 0.8

The statement which  is true for the quadrilateral is B.

To dilate a figure by a scale factor of 2, each point of the original figure is multiplied by 2.

So the coordinates of each vertex of A'B'C'D' are twice the coordinates of the corresponding vertex of ABCD.

The coordinates of A' are (4,10), B' are (4,4), C' are (8,6), and D' are (8,12).

To determine which statements are true, we can compare the angles and side lengths of the two quadrilaterals:

A) None of the answers apply. This may be a valid answer, but we should check the other options before concluding that none of them apply.

B) The angles of Quadrilateral ABCD and Quadrilateral A'B'C'D' are the same. This is true because dilation does not change angles. The corresponding angles of the two quadrilaterals are congruent.

C) The side lengths of Quadrilateral ABCD and Quadrilateral A'B'C'D' are not the same. We can see this by calculating the length of each side of both quadrilaterals.

Therefore, the correct answer is B.

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The analysis of the bank statement thus, given below. Since the result is negative, this means that Andre would still have a negative balance after depositing $100, and therefore would still be in debt.

1. If we put withdrawals and deposits in the same column, they can be represented as positive and negative values in a single column. Deposits would be represented with positive values, and withdrawals would be represented with negative values.

2. Andre's new balance would be $16.37. We can calculate this by subtracting $40 (the withdrawal) from his previous balance of $56.37:

$56.37 - $40 = $16.37

3. If Andre deposits $100 in this account, he will no longer be in debt. We can calculate his new balance by adding his previous balance and the deposit, and then subtracting any withdrawals:

$56.37 + $100 = $156.37 (balance after the deposit)

$156.37 - $28.50 - $37.91 - $16.43 - $42.00 - $72.50 - $45.00 - $50.00 - $10.03 - $62.47 = -$49.47

Since the result is negative, this means that Andre would still have a negative balance after depositing $100, and therefore would still be in debt.

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According to the information, he test statistic for this sample is 5.

To calculate the test statistic for the Wilcoxon signed-rank test, we need to calculate the differences between the "before" and "after" engagement ratings and rank them in order of their absolute values.

Student Before After Difference Absolute Difference Rank

1 30 40 10 10 1

2 20 40 20 20 2

3 32 37 5 5 3

4 43 46 3 3 4

5 48 44 -4 4 5

The sum of the ranks for the positive differences is 1 + 2 + 3 + 4 = 10, and the sum of the ranks for the negative differences is 5.

The smaller of the two sums (in this case, the sum of the ranks for the negative differences) is the test statistic for the Wilcoxon signed-rank test.

Therefore, the test statistic for this sample is 5.

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The equivalent ratio of dollars to pizza for 4 pizzas is 12 : 1.

A ratio is a comparison of two or more quantities that are related to each other in some way. It is expressed as a fraction or using the "colon" notation.

If Mike pays 12 dollars for one pizza, the ratio of dollars to pizza is:

12 : 1

To find the equivalent ratio for 4 pizzas, we need to keep the ratio of dollars to pizza constant. We can do this by multiplying both the numerator and denominator of the ratio by 4, since we are now dealing with 4 pizzas instead of 1. This gives us:

12 x 4 : 1 x 4

Simplifying this ratio gives us:

48 : 4

We can further simplify this ratio by dividing both the numerator and denominator by 4, which gives us:

12 : 1

Therefore, the equivalent ratio of dollars to pizza for 4 pizzas is 12 : 1.

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The simplified expression is 4(x - 3y).

logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if bx = n, in which case one writes x = logb n. For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log2 8.

Using the following logarithmic identities:

log a (bc) = log a (b) + log a (c)

log a (b/c) = log a (b) - log a (c)

We can simplify the expression as follows:

2㏒4 x - 6 log4 y = 2(㏒4) x - 6(㏒4) y

= 2(㏒4) x - 2(㏒4)3 y

= 2(㏒4)(x - 3y)

Now, we can simplify further by using the fact that ㏒4 = 2:

2(㏒4)(x - 3y) = 2(2)(x - 3y) = 4(x - 3y)

Therefore, the simplified expression is 4(x - 3y).

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