TELL WHETHER THE TRIANGLE IS A RIGHT TRIANGLE

Chelsea needs at least 190 cm² of paper to cover the box.

To find the minimum amount of paper Chelsea needs to cover the rectangular prism-shaped box, we need to calculate the surface area of the box.

Surface Area = 2(lw + lh + wh)

Where,

L is length, W is width, aH nd f f is height.

So, to find the minimum amount of paper Chelsea needs, we need to know the box's surface area of the box. Once we have the dimensions, we can plug them into the formula and calculate the surface area.

For example, if the box has dimensions of length of 10 cm, width 5 cm, and height 30 cm, the surface area would be:

Surface Area = 2(50 + 30 + 15)

Surface Area = 2(95)

Surface Area = 190 cm²

Therefore, Chelsea needs at least 190 cm² of paper to cover the box.

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The volume of a solid cone with a diameter of 10 centimeters and a height of 8 centimeters is equal to 209.47 cubic centimeters.

In Mathematics and Geometry, the volume of a cone can be determined by using this formula:

V = 1/3 × πr²h

Where:

Note: Radius = diameter/2 = 10/2 = 5 cm.

By substituting the given parameters into the formula for the volume of a cone, we have the following;

Volume of cone, V = 1/3 × 3.142 × 5² × 8

Volume of cone, V = 1/3 × 3.142 × 25 × 8

Volume of cone, V = 209.47 cubic centimeters.

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Complete Question:

A solid cone with a diameter of 10 centimeters and a height of 8 centimeters. Find the the volume of this cone.

The probability of x' bar to be greater than 59 is equal to 0.1193.

What is Mean value?

Apart from mode and median, mean is the one of the measures of central tendency. When we do the average of given set of values, it is called mean. Here in this question we need to check if the data meets the criterion for a normal distribution. One of these criterion is that data should be symmetric & bell shaped and another one is that mean and median should be approximately equal. By adding the total values given in the datasheet and dividing the sum by total no of values we will get the value of mean.

Here, the mean is = 162 point, the standard deviation = 140 point and x is considered as a variable for household water use in the country

μ=162

σ=140

X:- household water use for country

i.e., approx. Normal

(b) n=50

    n>30

i.e., approx. normal

(c) P(X'>59) = 1 - P(X'≤59)

                  = 1- P((X'-μ)/(σ/√n)≤59-57/(12/√50))

                  = 1- P(z ≤ 1.178)

                  = 1-0.8807

    P(X'>59)= 0.1193

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The correct result for solving the radical equation √(2x - 7) = 9 is derive to be 2x - 7 = 81, which makes the variable x = 44

In mathematics, the symbol √ is used to represent or show that a number is a radical. Radical equation is defined as any equation containing a radical (√) symbol.

The radical symbol for the equation √(2x - 7) = 9 can be removed squaring both sides as follows;

[√(2x - 7)]² = 9²

(2x - 7)^(2/2) = 9 × 9

2x - 7 = 81

the value of the variable x can easily then be derived as follows:

2x = 81 + 7 {collect like terms}

2x = 88

x = 88/2 {divide through by 2}

x = 44

Therefore, removing the radical symbol by squaring both sides of the equation will result to 2x - 7 = 81, and the variable x = 44

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4 mugs are required to fill if David made 4/3 of a quart of fruit juice. In each mug, he held 1/3 of a quart as per the fractions.

Fruit juice quart = 4/3

Holdes of quart =  1/3

This can be calculated by using the division of fractions. The number of mugs that can be filled will be calculated by using the fraction equation of division of quart of juice and holdes.

Mathematically,

number of mugs  = 4/3 ÷ 1/3

number of mugs  = 4/3 × 3/1

number of mugs = 4

Therefore we can conclude that David will able to fill in 4 mugs.

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Answer:

4x + y = 13

Step-by-step explanation:

The original equation is: 4x + y - 2 = 0 --> y = -4x + 2

Equation 1: 4x - y = 13 --> y = 4x - 13. Since it does not have the same slope as the original equation, Equation 1 is not the answer.

Equation 2 turns into y = -4x + 13. -3 = -4(4) + 13 = -16 + 13 = -3. Since the equation passes through this point and has the same slope as the original equation, it fits the criteria for the problem.

The linear function defined in the table is given as follows:

y = 0.5x + 9.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

From the table, we get that the slope and the intercept are obtained as follows:

Hence the function is:

y = 0.5x + 9.

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The value of the integral is (11/15)√3.

First, we can simplify the integrand using the trigonometric substitution:

Let x = 2sinθ

Then dx = 2cosθ dθ and 4x^2 - x^4 = 4(2sinθ)^2 - (2sinθ)^4 = 4(4sin^2θ - sin^4θ) = 4sin^2θ(4 - sin^2θ)

Substituting these expressions into the integral, we have:

∫2^3 1x^3 √4x2 − x4 dx

= ∫sin⁡(θ=π/6)sin(θ=π/3) 8sin^3θ √(4sin^2θ)(4-sin^2θ) (2cosθ)dθ

= 16∫sin⁡(θ=π/6)sin(θ=π/3) sin^3θ cos^2θ dθ

We can use the identity sin^3θ = (1-cos^2θ)sinθ to simplify the integral further:

16∫sin⁡(θ=π/6)sin(θ=π/3) sinθ(1-cos^2θ)cos^2θ dθ

Now, we can make the substitution u = cosθ, du = -sinθ dθ, and use the table of integrals to evaluate the integral:

16 ∫u=-√3/2u=1/2 -u^2(1-u^2) du

= 16 [(-1/3)u^3 + (1/5)u^5]u=-√3/2u=1/2

= 16 [(-1/3)(-√3/2)^3 + (1/5)(-√3/2)^5 - (-1/3)(1/2)^3 + (1/5)(1/2)^5]

= 16 [(-√3/24) + (3√3/160) + (1/24) - (1/160)]

= 16 [(11√3/480)]

= (11/15)√3

Therefore, the value of the integral is (11/15)√3.

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Answer:

3.92

Step-by-step explanation:

To evaluate the expression 5.6t when t=0.7, we just substitute 0.7 for t and multiply:

5.6(0.7) = 3.92

Therefore, 5.6t when t is 0.7 equals 3.92. Good Luck!

Applying L'Hopital's rule, we get: limx→0 sin 3x / 4x = limx→0 3cos 3x / 4 = 3/4 Therefore, the limit of sin 3x / 4x as x approaches 0 is 3/4.

(a) lim(x→0) (sin(3x) / (4x))

To evaluate this limit, we can use L'Hôpital's Rule, which states that if the limit of the ratio of the derivatives of two functions exists, then the limit of the original functions also exists and is equal to the limit of the ratio of their derivatives.

Step 1: Take the derivative of the numerator and denominator with respect to x:
- Derivative of sin(3x) with respect to x: 3cos(3x)
- Derivative of 4x with respect to x: 4

Step 2: Rewrite the limit using the derivatives:
lim(x→0) (3cos(3x) / 4)

Step 3: Evaluate the limit by plugging in x = 0:
(3cos(3*0) / 4) = (3cos(0) / 4) = (3*1) / 4 = 3/4

So, the given limit is 3/4.

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We need to calculate the first four derivatives of f(x) at x=0, and use the general formula for the Maclaurin series: f(x) = f(0) + f'(0)x + (f''(0)/2!)x² + (f'''(0)/3!)x³ + (f''''(0)/4!)x⁴ + ...

To find the Maclaurin series through degree 4 of a function f(x), we can use the formula:

f(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + (f''''(0)/4!)x^4 + ...

Here, we are given that f(x) = 2, which means that f'(x) = f''(x) = f'''(x) = f''''(x) = 0 for all values of x. Therefore, the Maclaurin series for f(x) through degree 4 is:

f(x) = 2 + 0x + (0/2!)x^2 + (0/3!)x^3 + (0/4!)x^4
    = 2

In other words, the Maclaurin series for f(x) is simply the constant function 2, since all of the higher-order derivatives of f(x) are zero.

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A 5 × 3 matrix A has 3 pivot columns. Col A is a subspace of [tex]R^5[/tex] with dimension 3, and Nul A is not equal to [tex]R^2[/tex]; it has a dimension of 0.

Suppose a 5 × 3 matrix A has 3 pivot columns. A pivot column is a column in a matrix that has a leading 1 (pivot position) after performing row reduction. Having 3 pivot columns in matrix A means there are 3 linearly independent columns.
Now, let's consider the two parts of your question:
1. Is Col A = R^5?
Col A represents the column space of matrix A, which is the span of its linearly independent columns. Since A is a 5 × 3 matrix with 3 linearly independent columns, the dimension of Col A (the column space) is 3. Therefore, Col A is a subspace of [tex]R^5[/tex], but not equal to [tex]R^5[/tex].
2. Is Nul A = [tex]R^2[/tex]?
Nul A represents the null space of matrix A, which is the set of all solutions to the homogeneous system Ax = 0. The dimension of the null space called the nullity of A, is equal to the number of columns minus the number of pivot columns. In this case, nullity(A) = 3 (number of columns) - 3 (pivot columns) = 0. This means Nul A has a dimension of 0, not 2, and consists only of the zero vector. So, Nul A ≠ [tex]R^2[/tex].
To summarize, Col A is a subspace of [tex]R^5[/tex] with dimension 3, and Nul A is not equal to [tex]R^2[/tex]; it has a dimension of 0.

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The inflection points occur at x = 0 and x = 5.

To find the x-coordinates of the inflection points for the polynomial p(x) = [tex]x^5/20 - 5x^4/12[/tex] + 2022/π, follow these steps:

1. Compute the second derivative of p(x):
  First derivative: p'(x) =[tex](5x^4)/20 - (20x^3)/12[/tex]
  Second derivative: p''(x) = [tex](20x^3)/20 - (60x^2)/12 = x^3 - 5x^2[/tex]
2. Set the second derivative equal to zero and solve for x:
[tex]x^3 - 5x^2 = 0[/tex]
 [tex]x^2(x - 5) = 0[/tex]

3. Find the x-coordinates where the second derivative is zero:
  x = 0 and x = 5

These x-coordinates are the inflection points for the polynomial p(x) = [tex]x^5/20 - 5x^4/12[/tex] + 2022/π. So, the inflection points occur at x = 0 and x = 5.

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The values of the expressions are: 5x, -5, 18 - z , -5,  16/m, 4, 36n, 3k +4 , 2,

Recall that a mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation and grouping to help determine order of operations and other aspects of logical syntax.

1a. the product of 5 and a number x.

= 5*x = 5x

b  Evaluate when x = -1.

= 5*-1 = -5

2a   18 decreased by a number z

this implies 18 - z

b Evaluate when z = 23.

18-23 = -5

3a The quotient of 16 and a number m

= 16/m

b  Evaluate when m=4

this means 16/4 = 4

4. aThe product of 8 and twice a number n

= 18*2(n)

= 36n

b. Evaluate when n = 3

= 36*3 = 108

5.aThe sum of 3 times a number k and 4

= 3(k) + 4

= 3k +4

b. Evaluate k= -2

= 3*-2 + 4

-6+4 = 2

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The interval of convergence is (-18/5, 18/5).

To express f(x) as a power series, we first need to decompose it into partial fractions:

f(x) = 11/(x^2 - 7x - 18) = 11/[(x - 9)(x + 2)]

Using partial fractions, we can write:

11/[(x - 9)(x + 2)] = A/(x - 9) + B/(x + 2)

Multiplying both sides by the denominator (x - 9)(x + 2), we get:

11 = A(x + 2) + B(x - 9)

Setting x = 9, we get:

11 = A(9 + 2)

A = 1

Setting x = -2, we get:

11 = B(-2 - 9)

B = -1

Therefore, we have:

f(x) = 1/(x - 9) - 1/(x + 2)

Now, we can write the power series of each term using the formula for a geometric series:

1/(x - 9) = -1/18 (1 - x/9)^(-1) = -1/18 * sigma^n=0 to infinity (x/9)^n

1/(x + 2) = 1/11 (1 - x/(-2))^(-1) = 1/11 * sigma^n=0 to infinity (-x/2)^n

So, putting everything together, we get:

f(x) = 1/(x - 9) - 1/(x + 2) = -1/18 * sigma^n=0 to infinity (x/9)^n + 1/11 * sigma^n=0 to infinity (-x/2)^n

The interval of convergence can be found using the ratio test:

|a_n+1 / a_n| = |(-x/9)^(n+1) / (-x/9)^n| + |(-x/2)^(n+1) / (-x/2)^n|

= |x/9| + |x/2|

= (|x|/9) + (|x|/2)

For the series to converge, we need |a_n+1 / a_n| < 1. This happens when:

(|x|/9) + (|x|/2) < 1

Solving for |x|, we get:

|x| < 18/5

Therefore, the interval of convergence is (-18/5, 18/5).

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Answer:

The Correct answer for x is 7

The following Venn diagram represents the supplied information:

        Milk

        /   \

       /     \

      /       \

  Coffee     Tea

    / \      /   \

   /   \    /     \

  /     \  /       \

M & C    C   T       M & T

 (6)    (12)    (2)   (8)

a) To find how many people like at least one drink, we need to add up the number of people in each region of the Venn-diagram:

Milk: 20

Tea: 30

Coffee: 22

Milk and Coffee only: 6

Coffee and Tea only: 2

Milk and Tea only: 8

Milk, Coffee, and Tea: 12

Adding these up, we get:

20 + 30 + 22 + 6 + 2 + 8 + 12 = 100

So 100 people like at least one drink.

b) To find how many people like exactly one drink, we need to add up the number of people in the regions that are not shared by any other drink:

Milk only: (20 - 6 - 8) = 6

Tea only: (30 - 2 - 8) = 20

Coffee only: (22 - 12 - 2) = 8

Adding these up, we get:

6 + 20 + 8 = 34

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To evaluate the integral ∫1−1(|2x|+2) dx exactly using the technique of example 2 in the text, we first split the integral into two parts:

∫1−1(|2x|+2) dx = ∫1^0 (-2x + 2) dx + ∫0^-1 (2x + 2) dx

Next, we can simplify the absolute values:

∫1^0 (-2x + 2) dx + ∫0^-1 (2x + 2) dx = ∫1^0 (-2x + 2) dx + ∫0^-1 (-2x + 2) dx

Now we can integrate each part:

∫1^0 (-2x + 2) dx + ∫0^-1 (-2x + 2) dx = [-x^2 + 2x]1^0 + [-x^2 + 2x]0^-1

Simplifying further, we get:

[-1^2 + 2(1)] - [0^2 + 2(0)] + [0^2 + 2(0)] - [-(-1)^2 + 2(-1)]
= 1 + 1 + 1 = 3

Therefore, the exact value of the integral ∫1−1(|2x|+2) dx is 3.
To evaluate the integral ∫₁₋₁ (|2x|+2) dx, we first need to split the integral into two parts due to the absolute value function.

For x >= 0, |2x| = 2x, and for x < 0, |2x| = -2x.

Now, we split the integral into two parts:

∫₁₋₁ (|2x|+2) dx = ∫₀₋₁ (-2x+2) dx + ∫₁₀ (2x+2) dx.

Now, we evaluate each integral separately:

∫₀₋₁ (-2x+2) dx = [-x²+2x]₀₋₁ = (1-2)-(0) = -1.

∫₁₀ (2x+2) dx = [x²+2x]₁₀ = (1+2)-(0) = 3.

Finally, we add the two results together:

∫₁₋₁ (|2x|+2) dx = -1 + 3 = 2.

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To obtain g(x) the graph of f(x) should be shifted in the right direction by 5 units

We can see and compare the graphs of f(x) and g(x) to see this visually

Since f(x) = x to the power of 2, so the graph of f(x) will be a parabola which will have center at the origin and opens upwards

The graph of g(x) will also be a parabola but it will have center at x = 5

So, we just need to shift the graph of f(x) by 5 units in right direction to obtain g(x)

In the equation of f(x), we just have to replace x with (x- 5) and we will get

g(x) = (x-5)^2

So, this will be the equation of parabola that's identical to f(x)

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We can start by setting the two functions equal to each other and solving for x:

f(x) = g(x)

x^2 = (x-5)^2

Expanding the right-hand side:

x^2 = x^2 - 10x + 25

Simplifying:

10x = 25

x = 2.5

So, the two functions intersect at x = 2.5. To shift f(x) to obtain g(x), we need to move it 5 units to the right, since the vertex of g(x) is at x = 5, which is 5 units to the right of the vertex of f(x) at x = 0.

Therefore, to obtain g(x) from f(x), we need to replace x with x-5:

g(x) = f(x-5) = (x-5) ^2

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Answer:

72

Step-by-step explanation:

12x12 is 144

and 144/2 is 72

Answer:

72

Step-by-step explanation:

12 X 12 = 144

Then you divide the answer to 2 and you get 72

You're Welcome! :D

Step 1: Find the number of bronze members

40% of the gym's members are bronze members. Therefore, we need to find 40% of 4000. We can do this either by multiplying 4000 by 0.4 (40% as a decimal) or setting up a proportion. I will demonstrate the proportion method.

percent / 100 = part / whole

40 / 100 = x / 4000

---Cross multiply and solve algebraically.

100x = 160000

x = 1600 bronze members

Step 2: Find the number of silver members

Using the same methodology as last time, we can set up and solve a proportion to find the number of silver members.

percent / 100 = part / whole

25 / 100 = y / 4000

100y = 10000

y = 1000 silver members

Step 3: Find the number of gold members

Now that we know how many bronze and silver members the gym has, we can subtract those values from the total number of members to find the number of gold members.

4000 - bronze - silver = gold

4000 - 1600 - 1000 = gold

gold = 1400 members

Answer: 1400 gold members

ALTERNATIVE METHOD OF SOLVING

Alternatively, we could have used the given percents and only used one proportion. We know percents have to add up to 100. We are given 40% and 25%, which means the remaining percent is 35%. Therefore, 35% of the members are gold members. Just as we did for the silver and bronze members above, we can set up a proportion and solve algebraically.

percent / 100 = part / whole

35 / 100 = z / 4000

100z = 140000

z = 1400 gold members

Hope this helps!

A simple random sample obtained by

selecting names from a list of all members of a population in a way that allows only chance to determine who is selected. So, option(C) is correct choice.

In probability sampling, the probability of each member of the population being selected as a sample is greater than zero. In order to reach this result, the samples were obtained randomly. In simple random sampling (SRS), each sampling unit in the population has an equal chance of being included in the sample. Therefore, all possible models are equally selective. To select a simple example, you must type all the units in the inspector. When using random sampling, each base of the population has an equal probability of being selected (simple random sampling). This sample is said to be representative because the characteristics of the sample drawn are representative of the main population in all respects. Following are steps for follow by random sampling :

Hence, for random sampling option(c) is answer.

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Answer:

true

Step-by-step explanation:

Yes, the statement "0.5 miles = 2,640 feet" is true.
One mile is equal to 5,280 feet, so half a mile (0.5 miles) is equal to 2,640 feet.

Or

1 mile = 5280

1/2 = 0.5 / 5280 = 5280 / 2 =2640

In a sample of 8 years, the mean number of years that the mango tree does not give fruit is 4.96, the variance is 1.87, and the standard deviation is 1.37.

The mean, variance, and standard deviation for the number of years that a mango tree does not bear fruit in a sample of 8 different years, given a 38% chance of bearing fruit in a given year, can be calculated using probability theory and statistical formulas.

To begin, we can find the probability of the mango tree not bearing fruit in a given year, which is 1 - 0.38 = 0.62. Using this probability, we can construct a binomial distribution with n = 8 trials and p = 0.62 probability of success (not bearing fruit). The mean (expected value) of the distribution is given by μ = np = 8 x 0.62 = 4.96.

The variance of the distribution is given by the formula σ^2 = np(1-p), which in this case equals 8 x 0.62 x 0.38 = 1.87. Finally, the standard deviation of the distribution is the square root of the variance, which equals sqrt(1.87) = 1.37.

Therefore, the mean number of years that the mango tree does not bear fruit in a sample of 8 years is 4.96, the variance is 1.87, and the standard deviation is 1.37. This means that we can expect the mango tree to bear fruit approximately 3 times in the sample of 8 years.

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The given differential equation xy" + 2y' - xy = 0 has a regular singular point at x=0, and the indicial roots of the singularity differ by an integer.

To show that the indicial roots of the singularity differ by an integer, we need to use the Frobenius method to obtain a series solution about x=0. The differential equation can be written as:

x^2y" + 2xy' - x^2y = 0

Assuming a series solution of the form y(x) = ∑n=0∞ anxn+r, we can substitute this into the differential equation and simplify the terms to obtain a recurrence relation for the coefficients an:

n(n+r)an + (n+2)(r+1)an+1 = 0

To ensure that the series solution converges, we require that the coefficient an does not become zero for all values of n, except for a finite number of cases. This condition leads to the indicial equation:

r(r-1) + 2r = 0

which gives the two indicial roots:

r1 = 0, r2 = -2

Since the difference between the two roots is an integer (2), we have shown that the indicial roots of the singularity differ by an integer.

Using r1 = 0 as the dominant root, the series solution for y(x) can be written as:

y1(x) = c0 + c1x - (c1/4)x^2 + (c1/36)x^3 - (c1/576)x^4 + ...

Using the formula (23) in Section 6.3, we can find a second linearly independent solution y2(x) in terms of y1(x) as:

y2(x) = y1(x) ∫ (e^-∫P(x)dx / y1^2(x))dx

where P(x) = 2/x - x. After simplification, we get:

y2(x) = c2x^2 + c3x^3 + (2c1/9)x^4 + ...

Therefore, the general solution of the given differential equation on (0,0) can be written as:

y(x) = c1x - (c1/4)x^2 + (c1/36)x^3 - (c1/576)x^4 + c2x^2 + c3x^3 + (2c1/9)x^4 + ...

or, simplifying further:

y(x) = x[c1 + c2x + c3x^2] + c1x[1 - (1/4)x + (1/36)x^2 - (1/576)x^3] + ...

where c1, c2, and c3 are constants determined by the initial/boundary conditions.

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similar triangles by AAA, thus

[tex]\cfrac{XT}{WY}=\cfrac{XZ}{ZY}\implies \cfrac{8}{4}=\cfrac{XZ}{5}\implies \cfrac{40}{4}=XZ\implies \stackrel{ meters }{10}=XZ[/tex]

(a) The junction capacitance per unit area is approximately 1.75 x 10^-5 F/cm². (b) To double the junction capacitance, we need to increase the acceptor concentration by a factor of 4. In other words, we need to increase NA from 2 x 10^15 cm⁻³ to 8 x 10^15 cm⁻³.

(a) The junction capacitance per unit area can be calculated using the following formula:

C = sqrt((qε/NA)(ND/(NA+ND))×V)

Where:

Plugging in the values given in the question, we get:

C = sqrt((1.6 x 10^-19 C × 12.4 ε0 / (2 x 10^15 cm⁻³)) × (3 x 10^16 cm⁻³ / (2 x 10^15 cm⁻³ + 3 x 10^16 cm⁻³)) × 1.6 V)

C ≈ 1.75 x 10^-5 F/cm²

(b) To double the junction capacitance, we need to increase the acceptor concentration (NA) by a certain factor. We can use the following formula to calculate this factor:

F = (C2/C1)² × (NA1+ND)/(NA2+ND)

Where:

Plugging in the values from part (a) as C1 and NA1, and using C2 = 2C1, we get:

2C1 = sqrt((qε/NA1)(ND/(NA1+ND))×V) × 2

Squaring both sides and simplifying, we get:

NA2 = NA1 × 4

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Answer:36

Step-by-step explanation: 3/4, but take away the denominator. now we have 3. what's a multiple of 3 and 9? 36.

now divide 36 by the denominator, that's 9. so that's the answer. (i think)

the final answer is:
[tex]\int \frac{ x^3} { x-1} dx = \frac{1}{3} x^3 + \frac{1}{2} x^2 + x + ln|x-1| + c[/tex] (where c is the constant of integration)

To evaluate the integral ∫ [tex]x^3 / x-1[/tex]dx, we can use long division or partial fraction decomposition to simplify the integrand.

Using long division, we get:

[tex]\frac{x^3}{ (x-1)} = x^2 + x + 1 + \frac{1}{ x-1}[/tex]
So, we can rewrite the integral as:

[tex]\int (x^2 + x + 1 + \frac{1}{(x-1)} dx[/tex]

Integrating each term separately, we get:

[tex]\int x^2 dx + \int x dx + \int dx + \int (1/(x-1)) dx\\= (1/3) x^3 + (1/2) x^2 + x + ln|x-1| + c[/tex]

Thus, the final answer is:

[tex]\int \frac{ x^3} { x-1} dx = \frac{1}{3} x^3 + \frac{1}{2} x^2 + x + ln|x-1| + c[/tex] (where c is the constant of integration)

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(a) The probability that x > 1/7 is 4/7

(b) The probability that x < 1 + 7y is 1/9.

(a) To find the probability that x > 1/7, we need to integrate the joint density function over the region where x > 1/7 and y is between 0 and 1:

[tex]P(x > 1/7) = \int \int _{x > 1/7} p(x,y) dx dy[/tex]

[tex]= \int_{1/7}^1 \int _0^1 2/3 (x + 2y) dx dy (since p(x,y) = 2/3 (x + 2y)[/tex]for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 otherwise)

[tex]= (2/3) \int_{1/7}^1 (\int_0^1 x dx + 2 \int_0^1 y dx) dy[/tex]

[tex]= (2/3) \int_{1/7}^1 (1/2 + 2/2) dy[/tex]

[tex]= (2/3) \int _{1/7}^1 3/2 dy[/tex]

= (2/3) (1 - 1/14)

= 12/21

= 4/7

Therefore, the probability that x > 1/7 is 4/7.

(b) To find the probability that x < 1 + 7y, we need to integrate the joint density function over the region where x is between 0 and 1 + 7y and y is between 0 and 1:

[tex]P(x < 1 + 7y) = \int \int_{x < 1+7y} p(x,y) dx dy[/tex]

=[tex]\int_0^1 \int_0^{(x-1)/7} 2/3 (x + 2y) dy dx[/tex](since p(x,y) = 2/3 (x + 2y) for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 otherwise)

= [tex](2/3) \int_0^1 (\int_{7y+1}^1 x dy + 2 \int_0^y y dy) dx[/tex]

= [tex](2/3) \int_0^1 [(1/2 - 7/2y^2) - (7y/2 + 1/2)] dx[/tex]

= [tex](2/3) \int_0^1 (-6y^2/2 - 6y/2 + 1/2) dy[/tex]

=[tex](2/3) \int_0^1 (-3y^2 - 3y + 1/2) dy[/tex]

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

= -2/9 + 1/3

= 1/9

Therefore, the probability that x < 1 + 7y is 1/9.

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