3.48 Referring to Exercise 3.39, find
(a) f(y|2) for all values of y;
(b) P(Y = 0 | X = 2).
this is 3.39
3.39 From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. If X is the number of oranges and Y is the number of apples in the sample, find (a) the joint probability distribution of X and Y ; (b) P[(X, Y ) ∈ A], where A is the region that is given by {(x, y) | x + y ≤ 2}.

All solutions lie on the line y = (4/3)x.

To find the solution set of the equation, we solve for y in terms of x as follows:

4x - 3y = 0

4x = 3y

y = (4/3)x

Therefore, the solution set of the equation is:

{(x, y) | y = (4/3)x}

This is a dependent equation, as it can be written in the form of y = mx, where m = 4/3, which is the slope of the line. In other words, all solutions lie on the line y = (4/3)x.

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To find a basis for the set of vectors in R² on the line y = -3x. we'll follow these steps:

Step 1: Write the equation in parametric form.

The given equation is y = -3x.

We can rewrite this equation in parametric form as follows: x = t y = -3t

Step 2: Identify a vector that lies on the line.

Now that we have the parametric form, we can use it to find a vector that lies on the line.

A general vector on the line can be represented as: v(t) = (t, -3t)

Step 3: Form the basis using the vector.

To find the basis for the set of vectors in R² on the line, we can choose a non-zero value for the parameter 't'.

Let's choose t = 1: v(1) = (1, -3)

The basis for the set of vectors in R² on the line y = -3x is { (1, -3) }.

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Matrix representation of the linear transformation T with respect to the bases B and A.

[tex][T]BA = [[1] [2] [3] [-1]][/tex]

We need to find the matrix representation of the linear transformation T with respect to the bases B and A.

First, let's find the images of the basis vectors in A under T:

T(1) = 1 + 0 + 0 + 0 = 1

T(2) = 2 + 0 + 0 + 0 = 2

T(1 + 2) = 1 + 0 + 2 + 0 = 3

T(1 - 2) = 1 + 0 - 2 + 0 = -1

We can write these as column vectors:

[T(1)]B = [1]

[T(2)]B = [2]

[T(1+2)]B = [3]

[T(1-2)]B = [-1]

To find the matrix representation of T with respect to B and A, we form a matrix whose columns are the coordinate vectors of the images of the basis vectors in B.

[tex][T]BA = [[T(1)]B [T(2)]B [T(1+2)]B [T(1-2)]B]= [[1] [2] [3] [-1]][/tex]

To check our answer, we can apply T to an arbitrary vector in Ps and see if we get the same result by multiplying the matrix [T]BA with the coordinate vector of the same vector with respect to the basis A.

For example, let's apply T to the vector [tex]v = 3 + 2r - 4r^2 + s[/tex] in Ps:

[tex]T(v) = T(3 + 2r - 4r^2 + s) = (3 - 4) + 0 + (3 - 8) + 0 = -6[/tex]

To find the coordinate vector of v with respect to A, we solve the system of equations

3 = a + 2b + c + d

2 = b

-4 = 2c - d

1 = 2a + 3b - c + 6d

which gives us a = -3/2, b = 2, c = -3/2, d = -5/2, so

[tex][v]A = [-3/2 2 -3/2 -5/2]^T[/tex]

Now we can compute [T]BA[v]A and see if we get the same result as T(v):

[tex][T]BA[v]A = [[1 2 3 -1] [-3/2 4 -3/2 -5/2]] [3 2 -4 1]^T= [-6 0]^T[/tex]

So we get the same result, which confirms that our matrix representation [T]BA is correct.

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The taylor polynomials of degree n approximating 1/(2-2x) for x near 0 is P7(x)=(1/2)+(1/2)x+(1/2)x2+(1/2)x3+(1/2)x4+(1/2)x5+(1/2)x6+(1/2)x7

What is taylor polynomials?

An infinite sum of terms stated in terms of the function's derivatives at a single point is referred to as a Taylor series or Taylor expansion of a function. Near this point, the function and the sum of its Taylor series are equivalent for the majority of common functions. If the functional values and derivatives are identified at a single point, the Taylor series is used to calculate the value of the entire function at each point.

P(x)=1/(2-2x)

=(1/2)(1/(1-x))

=(1/2)(1+x+x2+x3+x4+x5+x6+x7+x8+.....)

for n =3 ,P3(x)=(1/2)+(1/2)x+(1/2)x2+(1/2)x3

for n =5 ,P5(x)=(1/2)+(1/2)x+(1/2)x2+(1/2)x3+(1/2)x4+(1/2)x5

for n =7 ,P7(x)=(1/2)+(1/2)x+(1/2)x2+(1/2)x3+(1/2)x4+(1/2)x5+(1/2)x6+(1/2)x7

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The left out money = 20- 18.25

= $1.75

Given that:

2 bottles of multivitamin = $3.75 x 2 = $7.5

1 bottle of vitamin D = $4.85

2 Vitamin C bottles = $2.95 x 2 = $5.9

The total = 7.5 + 4.85 +5.9 = $18.25

If $20 had to be spent,

The left out money = 20- 18.25

= $1.75

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The matrix of the linear transformation t : R4 → R3 defined by t(x1, x2, x3, x4) = (2x1 3x2 x4, 5x1 9x3 − x4, 4x1 2x2 − x3 7x4) is:

| 2 0 0 0 |

| 0 0 5 -1 |

| 4 8 -2 7 |

To determine the matrix of the linear transformation t : R4 → R3, we need to find the image of the standard basis vectors under the transformation t. The standard basis vectors of R4 are e1 = (1, 0, 0, 0), e2 = (0, 1, 0, 0), e3 = (0, 0, 1, 0), and e4 = (0, 0, 0, 1).

Applying the transformation t to each of these vectors, we get:

t(e1) = (2, 0, 4)

t(e2) = (0, 0, 8)

t(e3) = (0, 5, -2)

t(e4) = (0, -1, 7)

Thus, the matrix of the linear transformation t with respect to the standard bases of R4 and R3 is:

| 2 0 0 0 |

| 0 0 5 -1 |

| 4 8 -2 7 |

Each column of this matrix represents the image of the corresponding basis vector of R4, expressed as a linear combination of the basis vectors of R3.

Note that the matrix has 3 rows and 4 columns, reflecting the fact that the transformation maps R4 to R3.

The first column represents the image of the first basis vector e1, which is (2, 0, 4) in R3.

Similarly, the second, third, and fourth columns represent the images of the basis vectors e2, e3, and e4, respectively.

Therefore, the matrix of the linear transformation t : R4 → R3 defined by t(x1, x2, x3, x4) = (2x1 3x2 x4, 5x1 9x3 − x4, 4x1 2x2 − x3 7x4) is:

| 2 0 0 0 |

| 0 0 5 -1 |

| 4 8 -2 7 |

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To determine if the integral is convergent or divergent, we need to evaluate the given integral. Integers are a type of number in mathematics that include both positive and negative whole numbers, as well as zero.

Here's the integral you provided:
∫(from 0 to 3) [(30x^2 - 7x)/10] dx

First, let's simplify the integer and by dividing each term by 10:
∫(from 0 to 3) [3x^2 - (7/10)x] dx

Now, we need to find the antiderivative of the simplified integrand:

Antiderivative of 3x^2 is x^3, and the antiderivative of (7/10)x is (7/20)x^2.

So, the antiderivative of the integrand is:
x^3 - (7/20)x^2

Next, we'll evaluate the antiderivative at the limits of integration (0 and 3):
(x^3 - (7/20)x^2) | (from 0 to 3)
= (3^3 - (7/20)(3^2)) - (0^3 - (7/20)(0^2))
= (27 - (7/20)(9)) - (0 - 0)
= 27 - (63/20)

Now, since we got a finite value for the integral, we can conclude that the integral is convergent.

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Answer: D

Step-by-step explanation:

There are 120 strings of length four that can be formed using the letters abcde if repetitions are not allowed.

Since repetition is not allowed, we can use the counting principle to determine how many chains of four can be formed from the letters abcde.

The primary position has five choices (a, b, c, d, or e). For the second position, he has 4 choices (because he cannot use the letter he chose for the first position).

The third position has three choices and the fourth position has two choices.

Utilizing the increase guideline, able to multiply the number of choices for each position to urge the overall number of conceivable strings.

 5x4x3x2 = 120

So, if repetition is not allowed, there are 120 strings of length 4 that can be formed using the characters abcde. 

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The ordered pairs in S are {(4, 4), (5, 5)}, and the ordered pairs in S–1 are {(4, 4), (5, 4), (5, 5), (6, 5)}.

The relation S is defined as x S y ⇔ x | y, which means that x divides y.

Using this definition, we can determine which ordered pairs are in S:

(3, 4) is not in S, since 3 does not divide 4

(3, 5) is not in S, since 3 does not divide 5

(3, 6) is not in S, since 3 does not divide 6

(4, 4) is in S, since 4 divides 4

(4, 5) is not in S, since 4 does not divide 5

(4, 6) is not in S, since 4 does not divide 6

(5, 4) is not in S, since 5 does not divide 4

(5, 5) is in S, since 5 divides 5

(5, 6) is not in S, since 5 does not divide 6

Therefore, the ordered pairs in S are:

{(4, 4), (5, 5)}

The relation S–1 is the inverse of S. An ordered pair (a, b) is in S–1 if and only if (b, a) is in S. In other words, (a, b) is in S–1 if and only if b divides a.

Using this definition, we can determine which ordered pairs are in S–1

(4, 3) is not in S–1, since 4 does not divide 3

(5, 3) is not in S–1, since 5 does not divide 3

(6, 3) is not in S–1, since 6 does not divide 3

(4, 4) is in S–1, since 4 divides 4

(5, 4) is in S–1, since 5 divides 4

(6, 4) is not in S–1, since 6 does not divide 4

(4, 5) is not in S–1, since 4 does not divide 5

(5, 5) is in S–1, since 5 divides 5

(6, 5) is in S–1, since 6 divides 5

Therefore, the ordered pairs in S–1 are  {(4, 4), (5, 4), (5, 5), (6, 5)}

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The given question is incomplete, the complete question is:

Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the “divides” relation. That is, for all (x, y) ∈ A x B,

x S y ⇔ x | y.

State explicitly which ordered pairs are in S and S–1.

7. For sin(x) with x₀ = 0, the Taylor series is given by:
sin(x) = Σ((-1)^n * x^(2n+1))/(2n+1)!
n=0 to infinity

The radius of convergence for sin(x) is infinite.

9. For x with x₀ = 1, the Taylor series is given by:
x = Σ(x - 1)^n
n=0 to 1

The radius of convergence for this series is infinite.

10. For x with x₀ = -1, the Taylor series is given by:
x = Σ(x + 1)^n
n=0 to 1

The radius of convergence for this series is infinite.

13. For 1/(1-x) with x₀ = 2, the Taylor series is given by:
1/(1-x) = Σ(-1)^n * (x - 2)^n
n=0 to infinity

The radius of convergence for this series is 1.

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The price of the phone increased by 10% from its initial value c, as indicated by the formula c(1.10) for the previous month's price.

A 10% increase would mean adding 10% of c to c itself if the cost of the phone had been c dollars two months prior. One way to put this is as.

Price last month [tex]= c + 0.10c[/tex]

Simplifying this expression, we can factor out c to get:

Price last month [tex]= c(1 + 0.10)[/tex]

Further, streamlining allows us to assess the expression enclosed in brackets:

If the phone cost c dollars two months ago, then a 10% increase would be 0.1c dollars.

The total of the initial price and the increase, which is:

[tex]c + 0.1c[/tex]

Price last month [tex]= c(1.10)[/tex]

Therefore, The price of the phone increased by 10% from its initial value c, as indicated by the formula [tex]c(1.10)[/tex] for the previous month's price.

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The least common multiple of the given two integers is 24804834 if the product of two integers is 27 x 38 × 52 × 711 and their greatest common divisor is 23 x 34 x 5.

We can use the formula

LCM(a, b) = (a * b) / GCD(a, b)

where LCM(a, b) is the least common multiple of a and b, and GCD(a, b) is their greatest common divisor.

We are given that the product of the two integers is

27 x 38 x 52 x 711

We can factor this into its prime factors

27 x 38 x 52 x 711 = 3^3 x 2 x 19 x 2^2 x 13 x 3 x 59 x 79

The greatest common divisor of the two integers is

23 x 34 x 5 = 2^2 x 5 x 23 x 17

We can now use the formula to find the least common multiple

LCM = (27 x 38 x 52 x 711) / (23 x 34 x 5)

LCM = (3^3 x 2 x 19 x 2^2 x 13 x 3 x 59 x 79) / (2^2 x 5 x 23 x 17)

Simplifying, we can cancel out the common factors of 2, 5, 23, and 3

LCM = 3^2 x 2 x 19 x 13 x 59 x 79 x 17

LCM = 24804834

Therefore, the least common multiple of the two integers is 24804834.

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The length of side VW is equal to 37 units.

In Mathematics and Geometry, the perimeter of a rectangle can be calculated by using this mathematical equation (formula);

P = 2(L + W)

Where:

By substituting the given side lengths into the formula for the perimeter of a rectangle, we have the following;

P = 2(4z + 1 + 3z + 2)

132 = 2(7z + 3)

132 = 14z + 6

z = 126/14

z = 9

VW = 4z + 1 = 4(9) + 1 = 37 units.

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The area under the standard normal curve between z = -1.15 and z = 2.84 is 0.8726.

To find the area under the standard normal curve between z = -1.15 and z = 2.84, we need to use a standard normal distribution table or a calculator.

Alternatively, we can use a software program such as R or Python to find the area.

Using a standard normal distribution table, we can find the areas to the left of z = -1.15 and z = 2.84, and then subtract the smaller area from the larger area to find the area between the two z-values.

From the table, we find:

The area to the left of z = -1.15 is 0.1251

The area to the left of z = 2.84 is 0.9977

Therefore, the area between z = -1.15 and z = 2.84 is:

0.9977 - 0.1251 = 0.8726

Rounding this to four decimal places, we get the final answer of 0.8726. Therefore, the area under the standard normal curve between z = -1.15 and z = 2.84 is 0.8726.

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The equation is  h = [tex]\frac{2A}{a+c}[/tex].

What is equation?

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

Here the given equation is ,

=> A = [tex]\frac{1}{2}[/tex](a+c)h

Now simplifying the equation then,

=> 2A = (a+c)h

=> h = [tex]\frac{2A}{a+c}[/tex]

Hence the equation is  h = [tex]\frac{2A}{a+c}[/tex].

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Answer:

The cost for Thato's usage of 35 kl of water is R702.48.

Step-by-step explanation:

Since Thato used 35 kl of water, she falls into the third category where the rate is R16.20 per kl. We can calculate the cost as follows:

Cost = Fixed charge + (Rate per kl × Usage) + Infrastructure fee

The fixed charge is free for infrastructure, so we don't need to include it in this calculation.

Cost = (Rate per kl × Usage) + Infrastructure fee

= (R16.20 × 35) + R7.15

= R567.00 + R7.15

= R574.15

We also need to add 15% VAT to the cost:

Total cost = Cost × (1 + VAT)

= R574.15 × 1.15

= R702.48

Therefore, the cost for Thato's usage of 35 kl of water is R702.48.

2.4.2. Answer: The new fixed charge would be R92.81.

Explanation:

If the fixed charge is increased by 15%, the new fixed charge would be:

New fixed charge = Old fixed charge + (15% of old fixed charge)

= R80.70 + (0.15 × R80.70)

= R80.70 + R12.11

= R92.81

Therefore, the new fixed charge would be R92.81.

We can express the number 0.94 as a ratio of integers by recognizing the repeating pattern in its decimal expansion and converting it to a fraction with a denominator of 100. The resulting fraction is 94/100, which simplifies to 47/50.

To express the number 0.94 as a ratio of integers, we need to find a pattern in its decimal expansion. As we can see, the decimal expansion of 0.94 repeats after the second digit, with the repeating pattern of 94. Therefore, we can write 0.94 as 94/100 or simplified to 47/50.

To understand this concept further, we can think of decimals as a shorthand way of writing fractions. A decimal is just another way to write a fraction with a denominator of 10, 100, 1000, etc. For example, 0.5 is equivalent to 5/10 or simplified to 1/2. In the case of 0.94, we can see that it is equal to 94/100, which can be further simplified to 47/50 by dividing both the numerator and denominator by 2.

The process of converting a decimal to a fraction can be useful in many different areas of math, including algebra, geometry, and calculus. It is important to understand this concept because fractions are an essential part of math and are used in many real-life situations, such as cooking, budgeting, and measurement.

In summary, we can express the number 0.94 as a ratio of integers by recognizing the repeating pattern in its decimal expansion and converting it to a fraction with a denominator of 100. The resulting fraction is 94/100, which simplifies to 47/50.

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Step-by-step explanation:

y'' = sinx

y' = -cosx + k

y = -sinx  + kx + c       if  y(0) = 0    then  c = 0

y = - sin x  + kx    Where k is a constant

Answer:

it is 140

Step-by-step explanation:

Step-by-step explanation:

y = (x^2+4x)    -2      take 1/2 of the x coefficient (4)  square it and add it and subtract it

y = ( x^2 + 4x +4 )  -4  -3     reduce everything

y = ( x+2)^2  - 7     Done.

Let A and P be square matrices,

D. det(PAP-1) = [(det P) (det A) (det P-1)]-1.

we can use the property of determinants that states det(AB) = det(A)det(B) for any matrices A and B.

We can rewrite PAP-1 as (P-1)-1APP-1, and then use the property of determinants to get:

det(PAP-1) = det((P-1)-1APP-1)
det(PAP-1) = det(P-1)-1det(A)det(P-1)

Since P is invertible, det(P) ≠ 0 and we can multiply both sides of this equation by det(P) to get:

det(P)det(PAP-1) = det(A)det(P-1)det(P)

Using the property of determinants again, we can simplify this equation to:

det(PAP-1) = det(A)det(P-1)

Finally, we can substitute det(P-1) = 1/det(P) into this equation to get:

det(PAP-1) = det(A)(1/det(P))
det(PAP-1) = (det(A)/det(P))

Therefore, the correct answer is D. det(PAP-1) = [(det P) (det A) (det P-1)]-1.

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Using the given data and a significance level of 0.01, the p-value for the test of the hypothesis is approximately 0.0151.

To calculate the p-value using Excel, we can first find the test statistic, which follows a t-distribution with degrees of freedom calculated using the formula:

df = (s1^2/n1 + s2^2/n2)^2 / [ (s1^2/n1)^2 / (n1-1) + (s2^2/n2)^2 / (n2-1) ]

where s1 and s2 are the population standard deviations, and n1 and n2 are the sample sizes.

Using the given values, we find that the degrees of freedom are approximately 86.9. Next, we can calculate the test statistic using the formula:

t = (x1-bar - x2-bar) / sqrt(s1^2/n1 + s2^2/n2)

which gives us a value of approximately -1.906. Finally, we can find the p-value using the Excel function T.DIST.RT, which calculates the right-tailed probability of a t-distribution. The formula for the p-value is:

p-value = T.DIST.RT(t, df)

Using Excel, we can enter the formula =T.DIST.RT(-1.906, 86.9) to find that the p-value is approximately 0.0151.

In conclusion, based on the given data and a significance level of 0.01, we can reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis that the true population mean of the first sample is less than the true population means of the second sample. The p-value of 0.0151 indicates that this conclusion is unlikely to be due to random chance alone.

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The derivative of the given function h(x) = 6x^(-4) - 3x^(-6) is h'(x) = -24x^(-5) + 18x^(-7).

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By using linear approximation formula the estimation g(1.99) and g(2.01) of  g(2) - 5 and g'(x) = Vx^2 + 5 are 4.91 and 5.09, respectively.

We can use the linear approximation formula, which is:

L(x) = f(a) + f'(a)(x-a)

Where L(x) is the linear approximation of f(x) at a,

f(a) is the value of f(x) at a, f'(a) is the derivative of f(x) at a, and x is the value we want to approximate.

In this case, we want to approximate g(1.99) and g(2.01) using the information given.

We know that g(2) = 5, so we can use a = 2 in the formula above.

We also know that g'(x) = Vx^2 + 5 for all x, so g'(2) = V(2)^2 + 5 = 9.

Therefore, we have:
L(1.99) = g(2) + g'(2)(1.99-2) = 5 + 9(-0.01) = 4.91
L(2.01) = g(2) + g'(2)(2.01-2) = 5 + 9(0.01) = 5.09

So the estimated values of g(1.99) and g(2.01) using linear approximation are 4.91 and 5.09, respectively, rounded to two decimal places.

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Var(X) (write it up to second decimal place) Var(X) is 1.98 (rounded to two decimal places).

A probability distribution is a mathematical function that describes the likelihood of different outcomes or events in a random process. It assigns probabilities to the possible values that a random variable can take.

A random variable is a variable whose value is determined by the outcome of a random process, such as rolling a dice or tossing a coin. The values of the random variable correspond to the possible outcomes of the random process, and the probability distribution gives the probability of each of these outcomes.

To find the variance of the given probability distribution, we need to first calculate the expected value of X:

μ = E(X) = ∑[xi * f(xi)] for all values xi in the distribution

μ = (10.1) + (20.4) + (30.15) + (40.25) + (5*0.1) = 2.65

Next, we can use the formula for variance:

Var(X) = E[(X - μ)^2] = ∑[ (xi - μ)^2 * f(xi) ] for all values xi in the distribution

Plugging in the values, we get:

Var(X) = (1-2.65)^20.1 + (2-2.65)^20.4 + (3-2.65)^20.15 + (4-2.65)^20.25 + (5-2.65)^2*0.1

Var(X) = 1.9825

Therefore, Var(X) is 1.98 (rounded to two decimal places).

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To calculate the residual for observation 6, we first need to find the forecast for observation 6. Based on the given information, the forecast for observation 6 is 34. Therefore, the residual for observation 6 would be:

Residual = Actual Demand - Forecast
Residual = 38 - 34
Residual = 4

So the residual for observation 6 is 4.
Hi! To find the residual for observation 6, we need to subtract the forecast (F) from the actual demand (A). In this case, the observation 6 values are:

Actual Demand (A): 38
Forecast (F): 28

Now, we'll calculate the residual:

Residual = Actual Demand (A) - Forecast (F)
Residual = 38 - 28
Residual = 10

So, the residual for observation 6 is 10.