1. (i) Find the first 3 terms in the expansion of (2 - y)^5 in ascending powers of y.
(ii) Use the result in part (i) to find the coefficient of x² in the expansion of (2 − (2x - x²))^5.

Answer:

We can use vectors to solve this problem. Let's assume that the position vector of point A is a, and the position vectors of points B and C are b and c respectively.

Since D is the midpoint of BC, we can find its position vector as:

d = (b + c)/2

Now, let's find the position vectors of M, N, and I using the given conditions:

MA + MB = 0

This means that the vector MA is equal in magnitude and opposite in direction to the vector MB. Since M is a point on the line segment AB, we can write:

MA = M - a

MB = M - b

So, MA + MB = 0 gives us:

M - a + M - b = 0

2M = a + b

M = (a + b)/2

Therefore, the position vector of M is:

m = (a + b)/2

Similarly, we can find the position vectors of N and I:

3NA + NC = 0

This means that the vector NA is three times the magnitude of the vector NC, and they are in opposite directions. Since N is a point on the line segment AC, we can write:

NA = N - a

NC = N - c

So, 3NA + NC = 0 gives us:

3(N - a) + (N - c) = 0

4N = 3a + c

N = (3a + c)/4

Therefore, the position vector of N is:

n = (3a + c)/4

IM + 2IN = 0

This means that the vector IM is twice the magnitude of the vector IN, and they are in opposite directions. Since I is a point on the line segment DM, we can write:

IM = I - m

IN = I - n

So, IM + 2IN = 0 gives us:

I - m + 2(I - n) = 0

3I = 2m + 2n

I = (2m + 2n)/3

Therefore, the position vector of I is:

i = (2m + 2n)/3

So, the points M, N, and I are located at:

M = (a + b)/2

N = (3a + c)/4

I = (2m + 2n)/3

where:

m = (a + b)/2

n = (3a + c)/4

The radius of the pond should be modified to 19.72ft to be able to maintain the original volume of the pond.

The original shape of the pond is the shape of a cylinder and the formula tye can be used to calculate the volume of a cylinder = πr²h

The volume of the original pond is calculated as follows;

π = 3.14

Radius = 18 ft

height = 36ft

Vol = 3.14×18×18×36

= 36,624.96ft³

The radius of the modified pond is calculated as follows; Volume = 36,624.96ft³

height = 30ft

radius = ?

That is;

36,624.96 = 3.14×r²× 30

r² = 36,624.96/94.2

= 388.8

r = √388.8

= 19.72 ft

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The calculated factored form of the expression x² + 6x - 16 is (x - 2)(x + 8)

From the question, we have the following parameters that can be used in our computation:

x² + 6x - 16

Expand the expression

So, we have

x² + 6x - 16 = x² + 8x - 2x - 16

Factorize the expression

So, we have the following representation

x² + 6x - 16 = x(x + 8) - 2(x + 8)

Factor out x + 8

So, the expression becomes

x² + 6x - 16 = (x - 2)(x + 8)

Hence, the factored form of the expression x² + 6x - 16 is (x - 2)(x + 8)

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The corresponding to Vector R is 4.8 units 47 East of South 4.8 units 47circ East of South

Vector C comprises a magnitude of -3.5i

Vector D is established as –3.3j (South bearing is thought to be negative along the y-axis).

Vector R = Vector D - Vector C

= (-3.3j) - (-3.5i)

= 3.5i - 3.3j

To evaluate the strength of Vector R, we must first compute its magnitude:

Magnitude of R = √((3.5)^2 + (-3.3)^2) ≈ 4.8 units.

determine the direction,

we shall need to calculate the angle θ with respect to the South direction (the negative y-axis):

tan(θ) = (3.5) / (3.3);

θ = arctan(3.5 / 3.3) ≈ 47°

Hence the answer is option 2

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Answer:

0

Step-by-step explanation:

The percent of fat calories an American consumes each day is normally distributed with a mean of 36 and a standard deviation of about 10.

Standard deviation is a measure of the spread of a data set. It is calculated by taking the square root of the variance of the data set. It is a measure of how much the individual values in a data set are spread out from the mean. It is used to quantify the amount of variation or dispersion of a set of data values.

According to the Central Limit Theorem, the mean of the sample X will be normally distributed with a mean of 36 and a standard deviation of 10/sqrt(20). This means that the mean percent of fat calories in the sample will be normally distributed with a mean of 36 and a standard deviation of about 3.16. This means that 68% of the time, the mean percent of fat calories in the sample will be between 32.84 (36-3.16) and 39.16 (36+3.16), 95% of the time it will be between 29.68 (36-6.32) and 42.32 (36+6.32), and 99% of the time it will be between 26.52 (36-9.48) and 45.48 (36+9.48).

In other words, this is happening because the Central Limit Theorem states that the sample mean will be normally distributed, and that the standard deviation of the sample mean will be the standard deviation of the population divided by the square root of the sample size. By understanding this theorem, we can calculate the probability that the mean of the sample will fall within certain ranges.

In conclusion, the percent of fat calories an American consumes each day is normally distributed with a mean of 36 and a standard deviation of about 10. By understanding the Central Limit Theorem, we can calculate the probability that the mean of the sample will fall within certain ranges. This allows us to make predictions about the sample mean and helps us understand why this is happening.

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The probability that the next bolt found would be non-defective is 0.98 or 98%.

A fundamental idea in mathematics and statistics, probability theory has several uses in the sciences, engineering, business, and social sciences. It is used to assess risks and uncertainties in diverse circumstances, analyse and forecast the possibility of events happening, and make decisions in the face of uncertainty.

Given:

Number of bolts already examined = 600

Number of defective bolts found = 12

Number of non-defective bolts = Total number of bolts examined - Number of defective bolts

= 600 - 12

= 588

Probability of finding a non-defective bolt = Number of non-defective bolts / Total number of bolts

= 588 / 600

= 0.98 or 98%

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Answer:

C

Step-by-step explanation:

When we do 30=y/3 we get 90 because of the fraction symbol and its says in the problem 3 times as much so more not less so 3x30 equals 90 so we already take out A and B and when there is nothing but a simple equation like 30=3 we dived so it cant be D Ethier so its C

PLS GIVE BRAINLIST

Happy Easter

Answer: f(x)= -0.333x + 2,333

The table for the population of the colony at the different times of the study is found.

Calculating the exponential growth as well as decay of a given collection of data is done using an exponential function, which is a mathematical function. Exponential functions, for illustration, can be used to estimate population changes, loan interest rates, bacterial growth, radioactive decay, as well as disease spread.

Given functions are-

P(t) = 9.[tex](1.02)^{t}[/tex]

Q(t) = 9.[tex]e^{0.02t}[/tex]

Population is estimated in thousands.

Table for the population of the colony:

t (time in months)     P(t) = 9.[tex](1.02)^{t}[/tex]                   Q(t) = 9.[tex]e^{0.02t}[/tex]

6                            P(6) = 9.[tex](1.02)^{6}[/tex] = 10.13         Q(t) = 9.[tex]e^{0.02*6}[/tex]= 10.14

12                          P(12) = 9.[tex](1.02)^{12}[/tex] = 11.41        Q(t) = 9.[tex]e^{0.02*12}[/tex] = 11.44

24                        P(24) = 9.[tex](1.02)^{24}[/tex] = 38.60     Q(t) = 9.[tex]e^{0.02*24}[/tex] = 14.54        

48                        P(48) = 9.[tex](1.02)^{48}[/tex]= 23.28      Q(t) = 9.[tex]e^{0.02*48}[/tex]= 23.50

100                    P(100) = 9.[tex](1.02)^{100}[/tex] = 65.20     Q(t) = 9.[tex]e^{0.02*100}[/tex]= 66.50

Thus, the table for the population of the colony at the different times of the study is found.

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The area of the field, given that it is a trapezium, would be 7, 986 m ² .

The field is in the shape of a trapezium which means that the area of the field can be found by the formula :

= ( a + b ) / 2 x height

Side a = 102 m

Side b = 140 m

height = 66 m

The area of the field is therefore:

= ( 102 + 140 ) / 2 x 66

= 121 x 66

= 7, 986 m ²

In conclusion, the area of the field is 7, 986 m ².

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The rate of interest according to the given data is 12.8%.

To answer this problem, we may utilize the compound interest formula:

A = P(1 + r/n)^(nt)

where A denotes the ultimate amount, P the principle, r the yearly interest rate, n the number of times the interest is compounded each year, and t the period in years.

We know that the compound interest on a certain payment over two years is Rs. 2600. Let us refer to the principal as P. Then we may type:

2600 = P(1 + r/100)^2 - P

When we simplify this equation, we get:

2600 = P[(1 + r/100)^2 - 1]

Dividing both sides by P[(1 + r/100)^2 - 1], we get:

1 = 1/[(1 + r/100)^2 - 1]

Simplifying even further, we get:

1 + (1 + r/100)^2 - 1 = (1 + r/100)^2

Taking the square root of both sides yields:

1 + r/100 = 1 + sqrt(2600/2500)

1 + r/100 = 1 + 0.04

r/100 = 0.04

r = 4%

As a result, the yearly interest rate is 4%. This, however, is the basic interest rate. To get the compound interest rate, use the following formula:

CI = P(1 + r/100)^2 - P

2500 = P(1 + 4/100)^2 - P

Simplifying this equation, we get:

2500 = P(1.04^2 - 1)

2500 = P(0.0816)

P = 30637.25

So the principal is Rs. 30637.25, and the annual compound interest rate is:

r = 100[(30637.25/10000)^(1/2) - 1]

r = 12.8%

Therefore, the rate of interest is 12.8% (rounded off to one decimal place).

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The z-scores of the two subjects are 1.18, and 1.25 respectively

Given data ,

The z-score is a measure of how many standard deviations a data point is away from the mean of a data set.

z = (X - μ) / σ

where X is the data point, μ is the mean, and σ is the standard deviation.

For the French exam:

X = 92 (score obtained by the student)

μ = 72 (mean of the French exam)

σ = 17 (standard deviation of the French exam)

On simplifying , we get

z_french = (92 - 72) / 17 = 1.18

For the math exam:

X = 80 (score obtained by the student)

μ = 70 (mean of the math exam)

σ = 8 (standard deviation of the math exam)

z_math = (80 - 70) / 8 = 1.25

Hence , the z-scores are calculated

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Answer:

8,11,14,17

Step-by-step explanation:

3n + 5

When n = 1,

3(1) + 5

=> 3 + 5

=> 8

When n = 2,

3(2) + 5

=> 11

When n = 3,

3(3) + 5

=> 14

When n = 4,

3(4) + 5

=> 17

The annual percentage rate for Terry's loan is 19.21%.

What is statistics?

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data.

To find the annual percentage rate (APR) for Terry's loan, we can use the following formula:

APR = (2 × Monthly Payment ÷ Loan Amount) × (12 ÷ Loan Term in Months + 1)

First, we need to calculate the total amount Terry will pay for the TV over the 12-month period:

Total payments = Monthly payment × Loan term in months

Total payments = $40.03 × 12

Total payments = $480.36

Next, we can use the formula to calculate the APR:

APR = (2 × $40.03 ÷ $450) × (12 ÷ 12 + 1)

APR = (0.1774) × (1.0833)

APR = 0.1921 or 19.21%

Therefore, the annual percentage rate for Terry's loan is 19.21%.

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The hypothesis test conclude that the training program was successful in increasing students' math skills.

As the average exam score of students who participated in the program was significantly higher than the population mean.

Number of students participated = 36

Population mean µ = 70

and σ = 12

To determine whether the math program was successful in increasing students' math skills, we can conduct a hypothesis test.

The null hypothesis denoted as H₀,

There is no difference between mean exam scores of students who participated in math program and population mean µ = 70.

The alternative hypothesis denoted as H₁,

There is a difference between the mean exam scores of the two groups.

Mathematically, the hypotheses as follows,

H₀: µ = 70

There is no difference in the mean exam scores of students who participated in the math program and the population mean.

H₁: µ > 70

The mean exam scores of students who participated in the math program are greater than the population mean.

Use a one-sample t-test to test the hypotheses.

The formula for the t-test statistic is,

t = (X - µ) / (s / √n)

X is the sample mean = 73.6

µ is the population mean = 70

s is the sample standard deviation

s = 12 / √36

  = 2

and n is the sample size= 36

Substituting these values into the formula, we get,

t = (73.6 - 70) / (2 / √36)

 = 3.6 / (1/3)

 ≈10.8

To determine whether the t-value is significant,

Compare it to the critical t-value at a chosen significance level α. Assuming significance level of 0.05 and 35 degrees of freedom df=n - 1 .

Critical t-value to be approximately 1.69.

Since calculated t-value 10.8 is greater than the critical t-value 1.69.

Reject the null hypothesis .

Therefore, using hypothesis test conclude that there is sufficient evidence to suggest that the math program was successful in increasing students' math skills.

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The mass of an "oxygen-molecule" is approximately 5.82 × 10⁴ times greater than the mass of an electron, and written in "standard-notation", the answer is 58287.59.

The "Standard-Notation" is defined as a way of writing a number using digits, a decimal point.

The mass of the "oxygen-molecule" is = 5.31×10⁻²³ grams,

The mass of the "electron" is = 9.11×10⁻²⁸ grams,

To find how many times greater the mass of an oxygen molecule is than the mass of an electron, we need to divide the mass of the oxygen molecule by the mass of the electron:

⇒ (5.31 × 10⁻²³)/(9.11 × 10⁻²⁸) = 58287.59 ≈ 5.82 × 10⁴.

Therefore, The mass of oxygen is 5.9 × 10⁴, times greater than electron.

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Check the picture below.

[tex]\textit{Law of sines} \\\\ \cfrac{\sin(\measuredangle A)}{a}=\cfrac{\sin(\measuredangle B)}{b}=\cfrac{\sin(\measuredangle C)}{c} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{\sin(54^o)}{31}=\cfrac{\sin(12^o)}{AB}\implies AB\sin(54^o)=31\sin(12^o) \\\\\\ AB=\cfrac{31\sin(12^o)}{\sin(54^o)}\implies AB\approx 8~in[/tex]

Make sure your calculator is in Degree mode.

The angles that is supplementary to <HAE is <IAE

Supplementary angles are simply described as a pair of angles whose sum is 180 degrees.

Some properties of supplementary angles are;

From the information given, we can see that;

<HAE is on a straight line with <IAE , making their sum to be 180 degrees

Then, <IAE is supplementary is <HAE

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Answer:

104 square meters

Step-by-step explanation:

Make sure all the units are same and consistent!!

The shape of the shaded region = Trapezium

h = Perpendicular height of trapezium

  = 16 m

∴Area of shaded region = Area of trapezium

                                          = [tex]\frac{1}{2}[/tex] × (Sum of parallel sides) × h

                                          = [tex]\frac{1}{2}[/tex] × [(9 + 4) m] × (16 m)

                                         = [tex]\frac{1}{2}[/tex] × (13 m) × (16 m)

∴Area of shaded region = 104 [tex]m^{2}[/tex]

                                         = 104 square meters

Okay, here are the steps to find the p-value:

1) The nationwide proportion of Americans who invest in the stock market is 55% (0.55)

2) The test statistic for Georgia is 1.75

3) To calculate the p-value, we need to know the degrees of freedom (df) and the critical value of the test statistic.

4) The df = 1, since we're comparing a single proportion (Georgia) to a fixed value (national average).

5) The critical value at df = 1 and 95% confidence is 3.84 (for a two-tailed test)

6) Since the test statistic of 1.75 is less than 3.84, we look up 1.75 in tables to find the p-value.

7) For a test statistic of 1.75 and df = 1, the p-value is 0.1860.

So the p-value to test if the proportion of Georgians who invest in the stock market is different than the nationwide average is 0.1860.

Let me know if you have any other questions!

Okay, here are the steps to find the p-value:

1) The nationwide proportion of Americans who invest in the stock market is 55% (0.55)

2) The test statistic for Georgia is 1.75

3) To calculate the p-value, we need to know the degrees of freedom (df) and the critical value of the test statistic.

4) The df = 1, since we're comparing a single proportion (Georgia) to a fixed value (national average).

5) The critical value at df = 1 and 95% confidence is 3.84 (for a two-tailed test)

6) Since the test statistic of 1.75 is less than 3.84, we look up 1.75 in tables to find the p-value.

7) For a test statistic of 1.75 and df = 1, the p-value is 0.1860.

So the p-value to test if the proportion of Georgians who invest in the stock market is different than the nationwide average is 0.1860.

Let me know if you have any other questions!

Answer:

Step-by-step explanation:

Since they are in a row, add all the measurements

4 ft 10 in

5 ft 6 in             add ft with the ft and in with in

3 ft 3 in

12 ft 19 in        

There are too many inches.  So you can take away 12 of those inches and and turn it into 1 more foot.  That would make 13 ft.  You'd be left with 7 in.

12 ft + 1 ft    19 in - 12 in

13 ft 7in

Answer:

Step-by-step explanation:

[tex]H_o: p = 0.5,H_1:p\neq 0.5[/tex]

[tex]p(\text{buttered side down})=\frac{19}{51}[/tex]

[tex]\pi _o=0.5\\n=51[/tex]

[tex]z=\frac{p-\pi _o}{\sqrt{\frac{\pi _o(1-\pi _o)}{n} } } =\frac{\frac{19}{51}-0.5 }\sqrt{{\frac{0.5(1-0.5)}{51} } }[/tex]

≈ [tex]-1.82[/tex]

a = 0.01    [tex]z_a/z=2.58\\|z| < |z_[ay}}_2|[/tex]

Thus, it cant reject H₀

The toast will land with the buttered side down 50% of the time.

{Hypothesis test}

The length of the arc, x, subtended by an angle of 74° in a circle with a radius of 8 inches is approximately 13.1 inches.

A central angle is an angle with endpoints located on a circle's circumference and a vertex located at the circle's center (Rhoad et al. 1984, p. 420). A central angle in a circle determines an arc.

The length of an arc, x, subtended by an angle of 74° in a circle with a radius of 8 inches can be found using the formula:

x = (θ/360) × 2πr

where θ is the central angle in degrees, r is the radius of the circle, and π is the mathematical constant pi.

x = (74/360) × 2π(8)

x ≈ 4.17π

To get the value to the nearest tenth of an inch, we can use the approximation π ≈ 3.14:

x ≈ 4.17 × 3.14

x ≈ 13.1

Therefore, the length of the arc, x, subtended by an angle of 74° in a circle with a radius of 8 inches is approximately 13.1 inches.

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Expanding the function expression f(x) = x^2 - 1, we get f(x) = (x - 1)(x + 1)

From the question, we have the following parameters that can be used in our computation:

f(x) = x^2 - 1

The above expression can be expanded using the difference of two squares

When this theorem is applied, we have

f(x) = (x - 1)(x + 1)

Hence, the function expression f(x) = x^2 - 1 when expanded is f(x) = (x - 1)(x + 1)

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Expanding the function expression f(x) = x^2 - 1, we get f(x) = (x - 1)(x + 1)

From the question, we have the following parameters that can be used in our computation:

f(x) = x^2 - 1

The above expression can be expanded using the difference of two squares

When this theorem is applied, we have

f(x) = (x - 1)(x + 1)

Hence, the function expression f(x) = x^2 - 1 when expanded is f(x) = (x - 1)(x + 1)

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