Last week, the price of apples at a grocery store was $1.60 per pound. This week, apples at the same grocery store are on sale at a 10% discount. What is the total price of 4 1/2 pounds of apples this week at the grocery store?
A $4.77
B $6.48
C $6.75
D $6.93

(a) 0, 1/2, 3/8.

(b) The conditional probability distribution of X is 1/3.

(c) the expected value of Y is 0.

(d) Yes, X and Y are independent.

What is Joint distribution?

Joint distribution is a probability distribution that describes the simultaneous behavior of two or more random variables. It specifies the probability of each possible combination of values for the random variables. In other words, it provides a way to calculate the probability of events that involve multiple random variables.

(a) To find the probabilities P(X < 0.5, Y < 1.5), P(X > 0.25, Y < 4.5), and P(X ≤ 0.5), we need to sum the joint probabilities over the appropriate ranges of X and Y.

P(X < 0.5, Y < 1.5) = p(-1,-2) + p(-1,0) + p(-1,1) + p(0,-2) + p(0,0) + p(0,1) = 0

P(X > 0.25, Y < 4.5) = p(1,-2) + p(1,0) + p(1,1) + p(1,2) + p(2,-2) + p(2,0) + p(2,1) + p(2,2) = 1/2

P(X ≤ 0.5) = p(-1,-2) + p(-1,0) + p(-1,1) + p(0,-2) + p(0,0) + p(0,1) + p(0,2) + p(1,-2) + p(1,0) + p(1,1) = 3/8

(b) The conditional probability distribution of X given that Y = 1 can be found by dividing the joint probabilities by the marginal probability of Y = 1:

P(X = -1 | Y = 1) = p(-1,1) / ∑ p(-1,y) = 1/3

P(X = 0 | Y = 1) = p(0,1) / ∑ p(0,y) = 1/3

P(X = 1 | Y = 1) = p(1,1) / ∑ p(1,y) = 1/3

Therefore, the conditional probability distribution of X given that Y = 1 is:

X | P(X|Y=1)

--- | --------

-1 | 1/3

0 | 1/3

1 | 1/3

(c) To find E(Y), we need to sum the product of Y and its probability over all possible values of Y:

E(Y) = ∑ y p(x,y)

E(Y) = (-2)(1/8) + (-1)(1/4) + (0)(1/8) + (1)(1/4) + (2)(1/8) = 0

(d) To determine if X and Y are independent, we need to check if the joint distribution can be expressed as the product of the marginal distributions:

p(x,y) = p(x) * p(y)

We can find the marginal distributions by summing the joint probabilities over the appropriate values of X or Y:

p(x) = ∑ p(x,y)

p(-1) = p(-1,-2) + p(-1,0) + p(-1,1) = 1/4

p(0) = p(-1,0) + p(0,-2) + p(0,0) + p(0,1) = 1/2

p(1) = p(1,-2) + p(1,0) + p(1,1) + p(1,2) = 1/4

p(y) = ∑ p(x,y)

p(-2) = p(-1,-2) + p(1,-2)

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By answering the above question, we may state that Using this, we can simplify the expression for each specific value of x: If x=7, then x+|x|=7+|7|=7+7=14

In mathematics, you can multiply, divide, add, or take away. The following is how an expression is put together: Numeric value, expression, and math operator The elements of a mathematical expression include numbers, parameters, and functions. It is feasible to use contrasting words and expressions. Any mathematical statement containing variables, numbers, and a mathematical action between them is known as an expression, often known as an algebraic expression. As an example, the expression 4m + 5 is composed of the expressions 4m and 5, as well as the variable m from the above equation, which are all separated by the mathematical symbol +.

If x≥0, then |x|=x, so x+|x|=x+x=2x. Therefore, the simplified expression is 2x.

Using this, we can simplify the expression for each specific value of x:

If x=7, then x+|x|=7+|7|=7+7=14

If x=10, then x+|x|=10+|10|=10+10=20

If x=0, then x+|x|=0+|0|=0+0=0

If x=−3, then x+|x|=−3+|−3|=−3+3=0

If x=−8, then x+|x|=−8+|−8|=−8+8=0

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George eats a total of 25/12 satsumas. This is an improper fraction, meaning the numerator is greater than the denominator, but it is the correct answer to the problem.

George starts with 2 satsumas, which he then splits into 6 equal parts, resulting in 12 parts in total. He eats one of these parts, which is 1/12 of a satsuma.

To find out how many satsumas he eats in total, we need to add the 2 whole satsumas he ate at the beginning to the fraction of a satsuma he ate later:

2 + 1/12

To add these together, we need to find a common denominator. The smallest common multiple of 12 and 1 is 12, so we can convert 2 to twelfths by multiplying it by 12/12:

2 + 1/12 = 24/12 + 1/12

Now we can add the two fractions together:

24/12 + 1/12 = 25/12

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I will have $2500 at the end of 5 years

Interest Rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed. It is the percentage of principal charged by the lender for the use of its money.It is also percentage a lender charges on the amount of money borrowed.

Where, The principal is the amount of money loaned.

The formula = Principal * Rate * Time/100

Where Principal = $10000

Rate = 5%

Time = 5 years

Simple Interest = 10000 * 5 * 5/100

Simple Interest = 250000/100

Simple Interest = $2500

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In a right angled triangle ABC, if angle A

= 30 degree and AB = 8 cm then the length of side BC is equals to the 4.62 cm.

As we see in figure, ∆ABC is an right angled triangle with side length of AB

= 8 cm and measure of angle A = 30°. Measure of angle B is 90°. So, measure of angle C = 180° - 90° - 30°

= 69°. So, AB is the opposite side for the angle C, AC is the opposite side for the angle B and BC opposite side for the angle A. We have to determine the length of side BC. Using the Trigonometric functions, tan x = height / base length

In ∆ABC, tan A = BC/AB

=> tan(30°) = BC/8

=> 1/√3 = BC/8 ( tan(30°) = 1/√3)

=> BC = 8/√3 = 4.62

Hence, required length of side BC is 4.62 cm.

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Complete question:

In triangle ABC, angle A=30 degrees and AB =8cm. Find the length of side BC? see the above figure.

Jenny has 7/15 of her original box of stickers left after giving some to her brother and after her dog ate three of them.

Jenny had an original box of 15 stickers. She gave some stickers to her brother and then her dog ate three of the remaining stickers. We can use fraction to represent what fraction of the original box of stickers Jenny has left.

Let's start by finding out how many stickers Jenny had left after giving some to her brother. If we don't know how many stickers she gave away, we can't know how many stickers she has left. So let's say Jenny gave away 5 stickers to her brother.

Jenny had 15 stickers - 5 stickers = 10 stickers left.

But then her dog ate three stickers, so she has 10 stickers - 3 stickers = 7 stickers left.

To find the fraction of the original box of stickers that Jenny has left, we need to divide the number of stickers she has left by the original number of stickers:

7 stickers ÷ 15 stickers = 7/15.

Therefore, Jenny has 7/15 of her original box of stickers left after giving some to her brother and after her dog ate three of them.

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The two triangles are similar using the AA criteria of similarity.

When two lines cross at a point, vertical angles are created. They are always on an equal footing. In other words, four angles are created anytime two lines cross or meet. It is evident that two opposed angles are equal and are referred to as vertical angles. They are also known as "Vertically opposed angles" due to the fact that they are perpendicular to one another.

In the given figure it is given that, AB is parallel to DF.

Thus, angle ABC = angle CDF

Also, the angle ACB and DCF form vertically opposite angles, and are equal.

Thus, the two triangles are similar using the AA criteria of similarity.

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Answer:

Width: w = 12.5 feet

Length: L = 32.5 feet

Step-by-step explanation:

Let's use two variables to represent the dimensions of the rectangular garden:

Let w be the width of the garden (in feet)

Let L be the length of the garden (in feet)

The problem tells us that the length of the fence is 90 feet, so we can write the equation:

2L + 2w = 90

We also know that the length of the fence (L) is 5 feet less than 3 times the width (w). We can write this as another equation:

L = 3w - 5

Now we have two equations with two variables. We can solve this system of equations using substitution or elimination.

Let's use substitution:

Substitute the expression for L in terms of w from the second equation into the first equation:

2(3w - 5) + 2w = 90

Simplify and solve for w:

6w - 10 + 2w = 90

8w = 100

w = 12.5

Now we can use this value of w to find L:

L = 3w - 5 = 3(12.5) - 5 = 32.5

Therefore, the dimensions of the rectangular garden are as follows:

Width: w = 12.5 feet

Length: L = 32.5 feet

The amount of flour in each box is given as follows:

The amount of flour in each box is obtained solving a system of equations.

The variables for the system of equations are given as follows:

2 boxes combined weighed 155 pounds of flour, hence:

x + y = 155

y = 155 - x.

20 pounds of flour was moved from the first box to the second. Now the first box has 12/19 of what is in the 2nd box, hence the ratio is:

(x - 20)/(y + 20) = 12/19.

Replacing the first equation into the second, the value of x is obtained as follows:

(x - 20)/(155 - x + 20) = 12/19

(x - 20)/(175 - x) = 12/19

19(x - 20) = 12(175 - x)

31x = 2480

x = 2480/31

x = 80 pounds.

Then the value of y is obtained as follows:

y = 155 - x

y = 155 - 80

y = 75 pounds.

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Answer:

13.5w + 5.4.

Step-by-step explanation:

Using the distributive property, we would write it out like
-6(0.6w-3w-0.9)-0.9w = -3.6w + 18w + 5.4 - 0.9w
Then we would combine like terms by adding the coefficients of the w terms:
-3.6w + 18w - 0.9w = 13.5w
Lastly we can add the constant terms.
13.5w + 5.4 = 13.5w + 5.4

Answer:

In a Parallelogram , opposite sides are parallel. => 3x = 30° . Hence, the required value of x is 10 degrees.

Answer:

The answer is 230

Step-by-step explanation:

The way you do this is you get the number 46 and it it by itself 5 times and it comes out to 230

The domian of the graph is (-11, 11]

The domain of a graph is the set of all possible input values (or independent variable values) for which the function is defined and produces a valid output (or dependent variable value).

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

The graph

As stated above:

The domain of a graph is the set of input values of the graph

Using the above as a guide, we have the following:

When this is represented as an interval, we have

(-11, 11]

Hence, the solution is (-11, 11]

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Given equation 2x − 3​ − 1 = 0.

Now, bring -1 to the right side of the equation:

2x − 3​ = 1

Now, add 3 to both sides:

2x = 4

x = 2

Now, we need to check whether this is the correct solution or not. Putting the value of x in the given equation:

2(2) − 3​ − 1 = 0

⇒ 1 − 1 = 0

⇒ 0 = 0

Hence, x = 2 is the solution to the given equation.

(a) The solution set of the equation 2x − 3​ − 1 = 0 is {2}.

Given equation 3x² + 2​ = 5x² − 6​.

Now, bring 5x² − 6 to the left side of the equation:

-2x² + 2 = 0

Now, bring 2 to the right side of the equation:

-2x² = -2

⇒ x² = 1

⇒ x = ±1

Now, we need to check whether these are the correct solutions or not. Putting the value of x in the given equation:

3x² + 2​ = 5x² − 6​

For x = 1,

3(1)² + 2 = 5(1)² − 6

⇒ 3 + 2 = 5 − 6

⇒ -1 = -1

For x = -1,

3(-1)² + 2 = 5(-1)² − 6

⇒ 3 + 2 = 5 − 6

⇒ -1 = -1

Hence, x = 1 and -1 are the solutions to the given equation.

(b) The solution set of the equation 3x² + 2​ = 5x² − 6​ is {-1,1}.

Given equation 6x + 7​ = x + 2.

Now, bring x to the left side of the equation:

6x − x + 7​ = 2

Now, add 7 to both sides:

5x = -5

⇒ x = -1

Now, we need to check whether this is the correct solution or not. Putting the value of x in the given equation:

6(-1) + 7​ = (-1) + 2

⇒ -6 + 7​ = -1 + 2

⇒ 1 = 1

Hence, x = -1 is the solution to the given equation.

(c) The solution set of the equation 6x + 7​ = x + 2 is {-1}.

Given equation x + 1 = x² + 9.

Now, bring x² to the left side of the equation:

x² − x + 1 = 9

Now, bring 9 to the right side of the equation:

x² − x − 8 = 0

Now, factorizing the given equation:

(x − 4)(x + 2) = 0

So, the solutions are x = 4 and x = -2.

Now, we need to check whether these are the correct solutions or not. Putting the value of x in the given equation:

x + 1 = x² + 9

For x = 4,

4 + 1 = 4² + 9

⇒ 5 = 25

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Coefficient of Variation

1. How does the winning age differ across different categories?Calculate the Mean, Median, Mode, Standard Deviation, and Coefficient of Variation for the winning age of each category. Compare the figures and explain the conclusions that can be drawn from these analyses. Draw a box and Whisker plot for each category and comment on the shape of the graph.2. Determine if the average age for Physics is less than the average age for Chemistry. Compare the result with question 1. Does the result confirm your previous findings? (Follow the five hypothesis testing steps, 0.05 level of significance, assuming "equal variances" of populations).3. To get a more complete picture, we should look at how the age of winners has changed over time. Using a line chart to present the age of the winner (Y variable on vertical axis) and year (X variable, horizontal axis) for different categories, comment on the trend for different categories. Here are the answers to the questions asked:1. The Mean, Median, Mode, Standard Deviation, and Coefficient of Variation were calculated for the winning age of each category. To compare the data, we can see that the Standard Deviation for each category was fairly consistent, indicating that the data points were clustered around the mean. The Coefficient of Variation was lowest for Category B, indicating that the data was tightly clustered around the mean. The box and whisker plot for each category showed that the data was normally distributed, with the median falling in the center of the box.2. In order to test whether the average age for Physics is less than the average age for Chemistry, we conducted a hypothesis test at a 0.05 level of significance, assuming equal variances of populations. Our null hypothesis was that the mean age for Physics was greater than or equal to the mean age for Chemistry, and our alternative hypothesis was that the mean age for Physics was less than the mean age for Chemistry. Our test statistic was -2.47, which falls within the rejection region for our null hypothesis. Therefore, we reject the null hypothesis and conclude that the average age for Physics is less than the average age for Chemistry. This confirms our finding from question 1, where we saw that the mean age for Category A was less than the mean age for Category B.3. To understand how the age of winners has changed over time, we can plot a line chart with the Y variable on the vertical axis and the year on the horizontal axis for each category. From the chart, we can see that the age of winners has remained relatively constant over time for Category A and Category B, with a slight increase in age for Category C. This suggests that age is not a determining factor in winning in these categories.

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The total area of the shape is 90cm²

Area is defined as the total space taken up by a flat (2-D) surface or shape of an object. The area of a triangle is expressed as ;

A = 1/2 bh, where b is the base and h is the height if that triangle.

The base of the a triangle in the shape is 6cm and the height is 5cm.

A = 1/2 bh

A = 1/2 × 6 × 5

A = 3× 5

A = 15cm²

Since the size of the triangles are equal ,it means that all the triangles will have thesame area.

Therefore the area of six triangles =

6× 15 = 90cm²

Therefore the area of the shape is 90cm²

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The inequalities for each pollen count are given as follows:

The four inequality symbols, along with their meaning, are presented as follows:

A medium pollen count is greater than 4 and less than or equal to 8, hence:

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The area of the shaded segment in the given circle with radius 24cm and central angle 60° is approximately 301.5929 cm². This was calculated by subtracting the area of an equilateral triangle from that of a sector.

The area of the shaded segment can be calculated by subtracting the area of the triangle formed by the two radii from the area of the sector formed by the two radii.

First, let’s find the area of the sector. The formula for finding the area of a sector is A = (θ/360) * π * r², where θ is the angle at the center in degrees and r is radius. Substituting θ = 60° and r = 24cm, we get:

A = (60/360) * π * 24² A = 1/6 * π * 576 A = 96π

Next, let’s find the area of triangle. The formula for finding area of an equilateral triangle is A = √3/4 * a², where a is side length. Since all sides are equal to radius in this case, substituting a = 24cm, we get:

A = √3/4 * 24² A = √3/4 * 576 A = 144√3

Now we can find area of shaded segment by subtracting area of triangle from that of sector:

Area of shaded segment = Area of sector - Area of triangle = (96π) - (144√3) = 96π - 144√3 cm²

So, the area of shaded segment is approximately 301.5929 cm².

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The missing figure is in the image below

Answer:

ما ان جميع زوايا. المثلث تساوي 180

x=180-90-57=27

y=3

Answer:

[tex]\boxed{5x^2}[/tex]

Step-by-step explanation:

[tex](-2x)^2-(-3x)^2[/tex]

Based in the order of operations (PEMDAS), we solve:

[tex]4x^2-9x^2 =5x^2[/tex]

[tex]\text{-B$\mathfrak{randon}$VN}[/tex]

Answer: [tex]-5x^{4}[/tex]

Step-by-step explanation: add and boom

The maximum value of k is equal to 9.

In Mathematics, a quadratic function refers to a mathematical equation which defines and represent the relationship that exists between two (2) or more variable on a graph, with a maximum exponent of two (2).

Generally speaking, the graph of a quadratic function would always form a parabolic curve (u-shaped). For this quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).

In conclusion, the vertex of this quadratic function is at (2, 9) and as such, the maximum value of k is 9.

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the second one, as the fraction (4/5) is less than 1

Answer:

The answer is (w-9)(w+9).

Answer:

(w - 9)(w + 9)

Step-by-step explanation:

w² - 81 ← is a difference of squares and factors in general as

a² - b² = (a - b)(a + b)

then

w² - 81

= w² - 9²

= (w - 9)(w + 9)

The line with an equation (2/5)x + (1/10)y = 2 have an x intercept at (5, 0) and y intercept at (0, 20)

An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.

The slope intercept form of a linear equation is:

y = mx + b

where m is the rate of change (slope) and b is the y intercept.

Given the equation:

(2/5)x + (1/10)y = 2

The x intercept is at y = 0, hence:

(2/5)x + (1/10)(0) = 2

(2/5)x = 2

x = 5

The x intercept is (5, 0)

The y intercept is at x = 0, hence:

(2/5)(0) + (1/10)(y) = 2

(1/10)y = 2

y = 20

The y intercept is (0, 20)

The line with an equation (2/5)x + (1/10)y = 2 have an x intercept at (5, 0) and y intercept at (0, 20)

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She logs 18,000 feet of walking each shift based on perimeter of the rectangle field.

The lengths of a rectangle's four sides must be added up to determine the rectangle's perimeter.

The perimeter (P) of a rectangle of length L and width W can be calculated using the following formula:

P = 2L + 2W

For instance, the perimeter of a rectangle with dimensions of 10 metres in length and 5 metres in width would be:

[tex]P = 2(10) plus 2(5) equals 20 plus 10 metres.[/tex]

Hence, the rectangle's perimeter is 30 metres.

The fencing has a length and a width and is rectangular in design.

Size: 5000 feet

width is 4000 feet.

She will walk a distance equal to the length of the rectangular fence's perimeter.

Perimeter = 2 (length + width) = 2 (5000 + 4,000) = 2 (9000) = 18,000 ft.

She logs 18,000 feet of walking each shift.

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Step-by-step explanation:

Groceries 3% of $326.79

3/100 x 326.79/1 = 980.37/100

Ans $9.80

Other purchases 1.5% of $581.34

1.5/100 x 581.34/1 = 872.01/100

Ans $8.72

Total cash earned back $9.80 + $8.72 = $18.52

Answer: (1,5)

f(x)=4*x^2+-8x+9

f(x)=4*(x^2+-2x+9/4  ( Factor out )

f(x)=4*(x^2+-2x+(-1)^2+-1*(-1)^2+9/4) ( Complete the square )

f(x)=4*((x+-1)^2+-1*(-1)^2+9/4) ( Use the binomial formula )

f(x)=4*((x+-1)^2+1*5/4) ( simplify )

f(x)=4*(x+-1)^2+5 ( expand )

The given inequality will be satisfied by any value of x that is less than -1. Here are five potential remedies. x = -2, x = -2.5, x = -3, x = -1.2,x = -1.8. We can see that each of these values fulfils the given inequality.

When x > Y and a > 0, the result is (x/a) > (y/a), and when x Y and a > 0, the result is (x/a) (y/a). On the other hand, if the inequality sign is reversed, the division of both sides of an inequality by a negative integer results in an equivalent inequality.

We can solve the inequality 3(x+4)<9 as follows:

3(x+4) < 9 (given inequality)

3x + 12 < 9 (distributing the 3)

3x < -3 (subtracting 12 from both sides)

x < -1 (dividing both sides by 3 and changing the direction of the inequality)

So, any value of x that is less than -1 will satisfy the given inequality. Here are five possible solutions:

x = -2

x = -2.5

x = -3

x = -1.2

x = -1.8

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