how many positive odd integers less than 10,000 can be written using digits 3,4,6,8, and 0

a) f(x) is a PDF. b) the marginal of X and Y is (y/2 + 1) / 2 c) the covariance of X and Y is: -1/18

What is meant by PDF?

In probability theory, a probability density function (PDF) is a function that describes the relative likelihood for a continuous random variable to take on a given value.

What is covariance?

Covariance is a statistical measure that quantifies the degree to which two random variables are linearly associated.

(a) To show that f(x) is a probability density function (PDF), we need to show that it satisfies the following two conditions:

Non-negativity: f(x) is non-negative for all x in its domain.

Normalization: The integral of f(x) over its domain is equal to 1.

The domain of f(x) is given by the inequality 0 ≤ x + 2y ≤ 2. To find the integral of f(x) over its domain, we need to integrate it with respect to y from (0-x/2) to (2-x/2), and then integrate the result with respect to x from 0 to 2:

∫(0 to 2) ∫(0-x/2 to 2-x/2) (x+y) dy dx

Solving the inner integral with respect to y, we get:

∫(0 to 2) [xy + [tex]y^2[/tex]/2] |_0-x/[tex]2^{(2-x/2)[/tex] dx

= ∫(0 to 2) ([tex]x^2[/tex]/4 - [tex]x^3[/tex]/12 + 1) dx

= [[tex]x^3[/tex]/12 - [tex]x^4[/tex]/48 + x] |_[tex]0^2[/tex]

= 2 - 2/3 + 2 = 8/3

Since the integral is finite and positive, the first condition of non-negativity is satisfied. To satisfy the normalization condition, we divide the function by the integral:

f(x) = (x+y) / (8/3)

Therefore, f(x) is a PDF.

(b) To find the marginal of X, we integrate f(x,y) over the range of y:

f(x) = ∫(0-x/2 to 2-x/2) (x+y) / (8/3) dy

= (x/2 + 1) / 2

Similarly, to find the marginal of Y, we integrate f(x,y) over the range of x:

f(y) = ∫(0 to 2) (x+y) / (8/3) dx

= (y/2 + 1) / 2

(c) To find the covariance of X and Y, we use the formula:

Cov(X, Y) = E[XY] - E[X]E[Y]

To find E[XY], we integrate xy*f(x,y) over the range of x and y:

E[XY] = ∫(0 to 2) ∫(0-x/2 to 2-x/2) xy*(x+y)/(8/3) dy dx

= ∫(0 to 2) [[tex]x^3[/tex]/6 - [tex]x^4[/tex]/24 + [tex]x^2[/tex]/4] dx

= 2/3

To find E[X] and E[Y], we integrate xf(x) and yf(y) over their respective ranges:

E[X] = ∫(0 to 2) x*(x/2+1)/2 dx

= 7/3

E[Y] = ∫(0 to 2) y*(y/2+1)/2 dy

= 7/6

Therefore, the covariance of X and Y is:

Cov(X, Y) = E[XY] - E[X]E[Y] = 2/3 - (7/3)*(7/6) = -1/18

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Given:

The number rounded to the nearest tenth is 59.6.

To find:

The approximate value of the given number to the nearest tenth.

Solution:

The required values shown in the below table:

Number               Nearest tenth                                    

59.58333333             59.6

The volume of the solid generated by the revolution is 33.5 cubic units.

It is given that, the diameter of the semi-circle is 4 and it is rotated to one full rotation around its diameter.

A solid generated when a semicircle is being rotated about its diameter is called a "SPHERE".

Therefore, the volume of the solid generated by the revolution is the volume of the sphere.

The formula for the volume of the sphere is given by,

Volume of sphere = (4/3)πr³

where r is the radius and π has the default value of 3.14

Here, the given diameter is 4.

To find the radius = diameter/2

radius = 4/2 = 2.

Now, to calculate the volume of the sphere substitute r=2 and π=3.14

volume of the sphere = (4/3)×3.14×2³

⇒ (4/3)×3.14×8

⇒ 100.48 / 3

⇒ 33.49 (approximately 33.5)

Therefore, the volume of the solid generated by the revolution is 33.5 cubic units.

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The measures of the angles in the unit circles are 53 degrees, 138 degrees, 300 degrees and 214 degrees

The question is an illustration of unit circles and the angles would be solved using

(x, y) = (cos(∅), sin(∅))

Figure (a)

Here, we have

(x, y) = (3, 4)

This means that

tan(∅) = y/x

So, we have

tan(∅) = 4/3

Take the arc tan of both sides

∅ = 53 degrees

Figure (b)

Here, we have

(x, y) = (-√5, 2)

This means that

tan(∅) = y/x

So, we have

tan(∅) = 2/-√5

Take the arc tan of both sides

∅ = 138 degrees

Figure (c)

Here, we have

(x, y) = (1, -√3)

This means that

tan(∅) = y/x

So, we have

tan(∅) = -√3/1

Take the arc tan of both sides

∅ = 300 degrees

Figure (d)

Here, we have

(x, y) = (-3, -2)

This means that

tan(∅) = y/x

So, we have

tan(∅) = -2/-3

Take the arc tan of both sides

∅ = 214 degrees

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Answer:

see explanation

Step-by-step explanation:

(a)

the sum of the interior angles of a polygon is

sum = 180° (n - 2) ← n is the number of sides

here n = 7 , then

sum = 180° × (7 - 2) = 180° × 5 = 900°

(b)

since the polygon is regular then the 7 interior angles are congruent

each interior angle = 900° ÷ 7 ≈ 128.6° ( to 1 decimal place )

(c)

the sum of the exterior angles of a polygon is 360°

since the polygon is regular then the 7 exterior angles are congruent

each exterior angle = 360° ÷ 7 ≈ 51.4° ( to 1 decimal place )

Answer:

a) 900°

b) ≈ 128,6°

c) ≈ 51,4°

Step-by-step explanation:

a) s = (n - 2) × 180°

n - the number of sides (in this case, it's 7)

s - the sum of interior angles

s = (7 - 2) × 180° = 5 × 180° = 900°

.

b) In order to find the size of each angle, we have to divide the sum by the number of sides:

900° / 7 ≈ 128,6°

.

c) Since the sum of exterior angles of a polygon is 360°, we can find the size of one exterior angle by dividing this sum by the number of sides:

360° / 7 ≈ 51,4°

If a 25 bag-sample has mean of 415 grams and variance of 400, then the p-value of test statistic is 0.0169.

To test whether the bag filling machine is working correctly at the 424 gram setting, we use one-sample t-test.

The null hypothesis is that mean weight of the bags filled by machine is 424 grams, and

The Alternative hypothesis is that the mean weight is less than 424 grams.

The test statistic "t" :

⇒ t = (415 - 424)/√(400/25)) = -2.25,

The degrees of freedom for this test is = 25-1=24.

To find the p-value of the test statistic, we need to determine the probability of observing a t-value less than or equal to -2.25,

Using a t-distribution table with 24 degrees of freedom, the p-value corresponding to t = -2.25 is approximately 0.0169.

Therefore, the p-value of the test statistic is 0.0169.

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What is triangle?

A triangle is a three-sided polygon with three vertices. The triangle's internal angle, which is 180 degrees, is constructed.

The given information is an acute angle ∠A, and sides a and b, along with the height of the triangle h = b sin A. The criteria for the number of possible triangles that can be formed are as follows:

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Answer:

b

Step-by-step explanation:

According to the puzzle the specified values are;

1. TU = 156.15m    

2. Area of rectangle = 12335.85 m²

3. GH = 182.33m

4. Area of triangle = 8934.17 m²

Three line segments meet at three non-collinear points to form a triangle, a geometric shape. The three points where the line segments meet are referred to as the vertices of the triangle, while the three line segments themselves are referred to as the sides of the triangle.

1. Given triangle TUV apply Pythagoras theorem,

175² = 79² + TU²   after simplification,

TU = 156.15m                   (option A is correct)

2. Area of rectangle = length × width

Area of rectangle = TU × UV  =  156.15 × 79  

Area of rectangle = 12335.85 m² (option G is correct)

3. Given triangle GHI apply Pythagoras theorem,

207² = 98² + GH²   after simplification,

GH = 182.33m    (option I is correct)

4. Area of triangle = [tex]\frac{1}{2}[/tex]  × base × height

Area of triangle =  [tex]\frac{1}{2}[/tex]  × HI × GH

Area of triangle =  [tex]\frac{1}{2}[/tex]  × 98m × 182.33m

Area of triangle = 8934.17 m²

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Tone's new monthly payment is 16.16% higher than his old monthly payment To solve the problem, we need to use the formula for the monthly payment of a mortgage loan:

monthly payment = (loan amount x monthly interest rate) / (1 - (1 + monthly interest rate)^(-n))

where:

loan amount = the amount of the loan

monthly interest rate = the annual interest rate divided by 12

n = the total number of payments (in months)

Let's first find the loan amount at the beginning of the mortgage. We know that Tone paid $9.99 per thousand, so we can set up a proportion:

$9.99 / 1000 = monthly payment / loan amount

Solving for the loan amount, we get:

loan amount = monthly payment / ($9.99 / 1000) = $1000 * (monthly payment / $9.99)

Substituting the given values, we get:

loan amount = $1000 * (monthly payment / $9.99) = $55,000

This means that Tone borrowed $55,000 at the beginning of the mortgage.

Now, let's find the old monthly payment. We know that Tone had a 10.5% adjustable rate mortgage for 30 years, which means he made 12 x 30 = 360 monthly payments. We can use the formula above to solve for the monthly payment:

monthly interest rate = 10.5% / 12 = 0.00875

n = 360

monthly payment = ($[tex]55,000 x 0.00875) / (1 - (1 + 0.00875)^(-360)[/tex]) = $512.47

Therefore, Tone's old monthly payment was $512.47.

Next, let's find the new monthly payment. We know that Tone will renew his mortgage at 12.5% interest rate, which means the new monthly interest rate is:

monthly interest rate = 12.5% / 12 = 0.01042

We also know that Tone will pay $10.68 per thousand, so we can set up a new proportion:

$10.68 / 1000 = new monthly payment / loan amount

Solving for the new monthly payment, we get:

new monthly payment = $10.68 / ($1000 / $55,000) = $595.40

Therefore, Tone's new monthly payment is $595.40.

Finally, let's find the percent increase in the new monthly payment compared to the old monthly payment. We can use the percent change formula:  

percent increase = ((new value - old value) / old value) x 100%

Substituting the given values, we get:

percent increase = (($595.40 - $512.47) / $512.47) x 100% = 16.16%

Therefore, Tone's new monthly payment is 16.16% higher than his old monthly payment

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Answer: y = 18000(0.89)^x, where x represents the number of years in the future.

Step-by-step explanation:

Answer:

Y= 18,000 X (0.11x)

Step-by-step explanation:

x being years y being car

The area of the square ABCD is approximately 420/π cm².

To find the area of the square, follow these steps:
1. Calculate the radius of the circle:
Since the area of the circle is 105 cm², you can use the formula for the area of a circle,

which is A = πr²,

where A is the area and r is the radius.
105 = πr²
Divide both sides by π:
r² = 105/π
Take the square root of both sides:
r = √(105/π)
2. Calculate the side length of the square:
Since the circle touches all 4 sides of the square, the diameter of the circle is equal to the side length of the square. The diameter is twice the radius (d = 2r), so:
d = 2√(105/π)
3. Calculate the area of the square:
Now, use the formula for the area of a square, which is A = s²,

where A is the area and s is the side length. In this case, s = d:
A = (2√(105/π))²
A = 4 * (105/π)
4. Multiply to find the area:
A = 420/π cm².

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The value of f(11) for the recursive rule and an explicit rule is 149.

An arithmetic sequence is a set of integers where each term is created by multiplying the preceding term by a fixed amount (referred to as the common difference). The nth term is often indicated by an, while the first term of the series is typically marked by a1. The following formula is used to determine the nth term in an arithmetic sequence:

a = a1 + (n-1)d

where d is the common distinction. In mathematics and other disciplines, arithmetic sequences are frequently employed to represent circumstances in which the pace or degree of change remains constant.

The recursive and explicit rule for the arithmetic sequence is given as:

an = a1 + (n-1)d

From the given graph f(1) = 19 and common difference = 32 - 19 = 13

The recursive rule is:

f(n) = f(n-1) + 13

The explicit rule is:

f(n) = 19 + 13(n-1)

Now, f(11) is calculated as:

f(11) = 19 + 13(11-1) = 19 + 130 = 149

Hence, the value of f(11) for the recursive rule and an explicit rule is 149.

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Answer: 90

Step-by-step explanation:

To find the median speed of the drives, list the speeds in order from lowest to highest and find the middle value. If there's an even number of speeds, find the average of the two in the middle.

To find the median speed of the drives in miles per hour, you would first arrange the speeds in order from smallest to largest and then identify the speed that sits exactly in the middle of this list. If there is an even number of observations, the median is the average of the two middle speeds.

For instance, if the speed of the drives listed in the table are 50 mph, 60 mph, 70 mph, 80 mph, and 90 mph, the median would be 70 mph. If the speeds were 50 mph, 70 mph, 80 mph, and 90 mph, the median would be the average of 70 and 80 mph, which is 75 mph.

The frequency table is useful in identifying how often each speed occurred, but it won't directly affect your calculation of the median. It's important to understand this concept as part of a broader understanding of statistics and data analysis.

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Answer:

Volume= 66.7π

Step-by-step explanation:

Volume = 1/3 ×π×h×r^2

V = 1/3 ×π ×8× 5 ^2

V = 66.7πinches^ 3

The expression that is equivalent to 5[4+3(x−6)] is 15x - 70

To simplify the expression  5[4+3(x−6)]  , we can first simplify the expression inside the square brackets, which is (x-6) multiplied by 3 and then added to 4. This gives us:4 + 3(x - 6) = 4 + 3x - 18 = 3x -

this expression back into the original equation, we get:5[4+3(x−6)] = 5(4 + 3x - 18) = 5(3x - 14) = 15x - 70

Therefore, the expression 15x - 70 is equivalent to 5[4+3(x−6)] as seen here.

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Missing parts;

Which expression is the equivalent to  5[4+3(x−6)]

Step-by-step explanation:

The given attachments are the steps to getting the answer...

Answer: 20+12=44-12

Step-by-step explanation:

You can distribute the multiplication on both sides of the equation.

Whenever you see a number before parentheses, it means to multiply that number to all terms in the parentheses.

so the new equation will be: (4 * 5) +(4 *3) = (2 *22) - (2*6)

Next, we simplify: 20 + 12 = 44 - 12

If the above equation is what you are looking for then use that but you can simplify further and add the numbers together so it will be

32=32  

Since the rectangle is made up of 2 congruent trapezoids, we can find the length and width of the rectangle by combining the lengths and widths of the trapezoids.

First, let's find the length of the rectangle:

The length of each trapezoid is the average of its bases:

length = (11 + 8) / 2 = 9.5

Since the trapezoids are congruent, the length of the rectangle is twice the length of a trapezoid:

length of rectangle = 2 * 9.5 = 19

Next, let's find the width of the rectangle:

The height of the trapezoids is the same as the height of the rectangle:

height = 6

The width of the rectangle is the same as the width of a trapezoid. To find the width of a trapezoid, we need to use the Pythagorean theorem, since the trapezoid has a height of 6 and bases of 11 and 8:

width = sqrt(6^2 + ((11-8)/2)^2) = sqrt(36 + 0.75) = sqrt(36.75)

So, the width of the rectangle is:

width = sqrt(36.75)

Finally, let's find the area of the rectangle:

area = length * width = 19 * sqrt(36.75) ≈ 87.83

Therefore, the length of the rectangle is 19 units, the width is approximately 6.07 units, and the area is approximately 87.83 square units.

Check the picture below.

If 4 vertices of a "regular-tetrahedron" are snipped off which leaves a "triangular-face" in place of each corner, then there will be 18 edges in the new polyhedron.

If we snip off the 4 vertices of a "regular-tetrahedron", leaving a triangular face in place of each corner and a hexagonal-face in place of each original face of the tetrahedron,

We get a truncated tetrahedron.

we know that a truncated-tetrahedron has ⇒ four regular "hexagonal-faces", four equilateral "triangular-faces", 12 vertices and 18 edges.

Therefore, the new polyhedron will have 18 edges.

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Step-by-step explanation:

Top floor is x     next floor is 2 x   then the bottom is 4 x

= 7x

7x = 245    x = 35

then  35 ft    70 ft   140 ft   = 245 ft total

The algebraic rule for each kind of rotation is completed below and a 360° rotation would appear as if the shape didn't rotate at all.

As a general rule of rotation, the algebraic rule for each kind of rotation are

Rotation Algebraic rule

90° clockwise about the origin (x, y)→ (y, -x)

180° about the origin (x, y) → (-x, -y)

270° clockwise about the origin (x, y) → (-y, x)

The images of the triangles cannot be drawn because they are not given

A 360° rotation about the origin would result in the original shape returning to its starting position.

It would appear as if the shape didn't rotate at all.

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Surface area of this right triangular prism=96in²

A right triangular prism is a three-dimensional geometric shape that has two parallel triangular bases and three rectangular faces that connect the bases. The triangular bases of a right triangular prism are always perpendicular to its rectangular faces. The prism is called "right" because the rectangular faces meet the bases at right angles, meaning the edges connecting the triangular and rectangular faces form right angles.

In the given  right triangular prism

S₁=5in

S₂=5in

S₃=8in

l=4in

b=8in

h=3in

Total surface area of this right triangular prism = (S₁+S₂+S₃)l+(b×h)

=(5+5+8)×4+8×3

=96

Hence, surface area of this right triangular prism=96in².

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y = (46.5 - (0.5b / 58))x + b This equation allows us to plug in any value for x (the number of miles driven) and calculate Kristen's total pay for that month.

To write an equation in slope intercept form to find Kristen's total pay, we need to first determine the variable values for the equation. In this case, we know that Kristen is paid the same salary each month, so we can assign a constant value to represent her base pay. Let's call this value "b" for simplicity.

Next, we need to determine the relationship between the number of miles Kristen drives and the extra pay she receives. We can do this by calculating the rate at which she is paid per mile. To do this, we can use the formula:

Pay per mile = (Total pay - Base pay) / Number of miles driven

Using the values given in the problem, we can calculate the pay per mile for March and April as follows:

March:

Pay per mile = ($7,748.00 - b) / 58 miles

April:

Pay per mile = ($8,532.00 - b) / 72 miles

To find the overall slope of the equation, we can calculate the difference in pay per mile between March and April:

Slope = (Pay per mile for April - Pay per mile for March) / (Number of miles driven in April - Number of miles driven in March)

Slope = (($8,532.00 - b) / 72 miles) - (($7,748.00 - b) / 58 miles)) / (72 miles - 58 miles)

Simplify the equation and solve for the slope:

Slope = 46.5 - (0.5b / 58)

Now that we have the value of the slope, we can write the equation in slope intercept form:

y = mx + b

y represents Kristen's total pay, m represents the slope we just calculated, x represents the number of miles she drives, and b represents her base pay.

Putting it all together, the final equation for Kristen's total pay in a month given the number of miles she drives is:

y = (46.5 - (0.5b / 58))x + b

This equation allows us to plug in any value for x (the number of miles driven) and calculate Kristen's total pay for that month.

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The probability of event A, the sum being greater than 5, is 11/36. The probability of event B, the sum being an odd number, is 1/2.

To calculate the probability of each event, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event.

Event A: The sum is greater than 5. There are 11 favorable outcomes (6,5), (6,4), (6,3), (6,2), (6,1), (5,6), (4,6), (3,6), (2,6), (1,6), and (5,5). The total number of possible outcomes is 36. So, the probability of event A is 11/36.

Event B: The sum is an odd number. There are 18 favorable outcomes (1,3), (1,5), (1,5), (2,1), (2,3), (2,5), (2,3), (3,1), (3,3), (3,5), (4,1), (4,3), (4,5), (5,1), (5,3), (5,5), (6,1), (6,3), and (6,5). The probability of event B is 18/36 or 1/2.

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The true statements about the image △A'B'C' include the following:

A. AB is parallel to A'B'.

B. [tex]D_{O, \frac{1}{2} }[/tex] (x, y) = (1/2x, 1/2y)

C. The distance from A' to the origin is half the distance from A to the origin.

In Mathematics and Geometry, a dilation simply refers to a type of transformation which typically changes the size of a geometric object, but not its shape.

This ultimately implies that, the size of the geometric shape would be increased (stretched or enlarged) or decreased (compressed or reduced) based on the scale factor applied.

Next, we would apply a dilation to the coordinates of the pre-image by using a scale factor of 0.5 centered at the origin as follows:

Ordered pair A (-4, 3) → Ordered pair A' (-4 × 1/2, 3 × 1/2) = Ordered pair B' (-2, 1.5).

Ordered pair B (4, 4) → Ordered pair B' (4 × 1/2, 4 × 1/2) = Ordered pair B' (2, 2).

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The probability that no two adjacent people would stand is option C: 49/256, as there are 256 distinct counting rotations.

First, we assume that one person stands up. There are 8 possible ways to select which person does. We can, next, assume that no two adjacent people stand up. There are 4 ways to choose which of the remaining 7 people will stand up (since no two adjacent people can stand up).

Then, we can assume that no two adjacent people stand up again. There are 3 ways to choose which of the remaining 6 people will stand up (since no two adjacent people can stand up).

We continue this process until there is only one person left who can stand up.

The number of arrangements according to the number of people standing.

Therefore, the number of arrangements that no two adjacent people will stand.

n(S) = 1 + 8 + 20 + 16 + 2 + 2 = 49

The total number of possible outcomes is:

2⁸ = 256

So, the probability that no two adjacent people will stand would be:

= 49/256.

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Complete question is:

Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?

(a) 47/256

(b) 3/16

(c) 49/256

(d) 25//128

(e) 51/256

The annual rate of decline is 5.3%.

What is exponential ?

Exponential refers to a mathematical function where the variable is in the exponent. Exponential functions have the general form:

[tex]f(x) = a^{x}[/tex]

here "a" is a constant called the base, and "x" is the variable. When the base "a" is a positive number greater than 1, the function grows exponentially as "x" increases. When the base "a" is a number between 0 and 1, the function decays exponentially as "x" increases.

We can start by using the formula for exponential decay:

[tex]N(t) = N0 * (1 - r)^{t}[/tex]

where N(t) is the population after t years, N0 is the initial population, r is the annual rate of decay (as a decimal), and t is the time in years.

We are given that the initial population is 670, so N0 = 670. After 4 years, the population has reduced to 557, so N(4) = 557. We can plug these values into the formula and solve for r:

[tex]557 = 670 * (1 - r)^{4}[/tex]

[tex](557/670)^{1/4} = 1 - r[/tex]

[tex]0.947 = 1 - r[/tex]

[tex]r = 0.053[/tex]

So the annual rate of decline is 5.3%.

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