How many degrees are in a semicircle? 36° 90° 270° 180°

The thirty-fourth term of the sequence is 1282.

First, we can find the second term of the ap series as follows:

a₂ = a₁ + 36 = a₄₃ + 36

Similarly, we can find the third term of the sequence as:

a₃ = a₂ + 36 = a₄₄ + 36

Continuing this pattern, we can find the thirty-fourth term of the sequence as:

a₃₄ = a₃₃ + 36 = (a₃₂ + 36) + 36 = ((a₃₁ + 36) + 36) + 36 = so on = a + 33(36)

Substituting the given value of a = 146, we get:

a₃₄ = 146 + 33(36) = 1282

a₃₄ = 1282

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The model that fits the data is a linear model and the equation is y = -72.5x + 2102.6

See attachment

Based on the graph, the model would you choose to fit this data is a linear regression, and the equation is

y = -72.5x + 2102.6

This is because the points appears to be linear

We have

x = 3

So, we have

y = -72.5 * 3 + 2102.6

y = 1885.1

So, the amount is 1885.1

Here, we have

y = 0

So, we have

-72.5x + 2102.6 = 0

This gives

x = 29

Hence, the time is 29 minutes

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Shannon broke down 1536 into three numbers: 256, 72, and 56, and then added the products of these three numbers to get her final answer of 256.

what is products and addition ?

The product is the result of multiplying two or more numbers together. For example, the product of 2 and 3 is 6, denoted as 2 x 3 = 6.Addition is another arithmetic operation that involves combining two or more numbers to find a total or a sum.

In the given question,

To solve 1536 ÷ 6, Shannon broke down the number 1536 into three numbers that are divisible by 6.

Let's start by finding the factors of 6: 1, 2, 3, and 6.

We need to find three numbers whose sum is equal to 1536 and each of which is divisible by 6.

We can start with the largest factor of 6, which is 6 itself. We can divide 1536 by 6 to get 256.

Now we need to find two more numbers whose sum is 128 and are also divisible by 6.

One such pair of numbers is 72 and 56.

We can verify that the sum of these three numbers is indeed 1536:

256 + 72 + 56 = 384 + 56 = 440 + 256 = 696 + 840 = 1536.

So, Shannon broke down 1536 into three numbers: 256, 72, and 56, and then added the products of these three numbers to get her final answer of 256.

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A) Surface area of Can is 276.32 [tex]cm^2[/tex].

B)The reason is that the top of the can has to be open to be used as a pencil holder. Therefore, you do not have to include the area of the top which is  [tex]\pi r^2[/tex] .

What is surface area?

A three-dimensional object's surface area is the space it takes up when viewed from the outside.

Here the given radius = 4 cm and height h = 9 cm.

A) Now using cylinder surface area formula then,

=> Surface area SA = [tex]\pi r^2+2\pi rh[/tex] square unit.

=> SA = [tex]3.14\times4^2+2\times3.14\times4\times9[/tex]

=> SA = [tex]3.14\times16[/tex]+ 226.08

=> SA = 50.24+226.08

=> SA = 276.32 [tex]cm^2[/tex]

B) Here top of the paint is used to open a pencil holder. So we don't need bottom circle area. [tex]\pi r^2[/tex] is the formula for the area of a circle which means you can use it to calculate the area of the top of the can.

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[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{-10}{h}~~,~~\underset{4}{k})}\qquad \stackrel{radius}{\underset{2}{r}} \\\\[-0.35em] ~\dotfill\\\\ ( ~~ x - (-10) ~~ )^2 ~~ + ~~ ( ~~ y-4 ~~ )^2~~ = ~~2^2\implies (x+10)^2+(y-4)^2=4[/tex]

Answer:

Step-by-step explanation:

the answer willbe 80

Each of the expressions has been simplified by using properties of exponents as shown below.

In Mathematics, an exponent refers to a mathematical operation that is typically used in conjunction with an algebraic expression in order to raise a quantity to the power of another.

This ultimately implies that, an exponent is represented by the following mathematical expression;

bⁿ

Where:

By applying the multiplication and division law of exponents for powers to each of the expressions, we have the following:

(1.25 × 10⁷) + 63,000,000 = (1.25 × 10⁷) + (6.3 × 10⁷) = (1.25 + 6.3) × 10⁷ = 7.55 × 10⁷

12,000 + 7 × 10⁴ = 1.2 × 10⁴ + 7 × 10⁴ = (1.2 + 7) × 10⁴ = 8.2 × 10⁴

5.88 × 10⁵ - 3.44 × 10⁵ = (5.88 - 3.44) × 10⁵ = 2.44 × 10⁵

6 × 10⁸ ÷ 120 = 6 × 10⁸ ÷ 1.2 × 10² = (6 ÷ 1.2) × 10⁸⁻² = 5 × 10⁶

9 × 10¹² ÷ 4.5 × 10⁴ = (9 ÷ 4.5) × 10¹²⁻⁴ = 2 × 10⁸

1.3 × 10³ · 2,200 = 1.3 × 10³ × 2.2 × 10³ = (1.3 × 2.2) × 10³⁺³ = 2.86 × 10⁶

(6.3 × 10⁴) + 1.3 × 10⁴ = (6.3 + 1.3) × 10⁴ = 7.6 × 10⁴

3,400 · 2 × 10⁴ = 3.4 × 10³ × 2 × 10⁴ = (3.4 × 2) × 10³⁺⁴ = 6.8 × 10⁷

9.78 × 10⁵ - 732,000 = (9.78 - 7.32) × 10⁵ = 2.46 × 10⁵

3.5 × 10² · 2 × 10⁴ = (3.5 × 2) × 10²⁺⁴ = 7 × 10⁶

1.1 × 10⁸ ÷ 22,000 = 1.1 × 10⁸ ÷ 2.2 × 10⁴ = (1.1 ÷ 2.2) × 10⁸⁻⁴ = 0.5 × 10⁴ = 5 × 10³.

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Answer:

the best move is king because it did not have a king

Applying the Poisson probability distribution​ formula,  the values of μ​, ​x, and e are: μ​ ≈ 0.866, ​x = 4, and e ≈ 2.71828

To apply the Poisson distribution formula, we need to identify the values of μ​, ​x, and e:

μ​ is the mean or average number of hits per region. We can calculate μ​ as the total number of hits divided by the total number of regions:

μ​ = total number of hits / total number of regions = 505 / 583

μ ≈ 0.866

​x is the number of hits we want to find the probability for, which in this case is 4.

e is the mathematical constant e, which is approximately equal to 2.71828 and is used in the Poisson distribution formula to calculate the probability of x hits.

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Consider the events , , . if a player rolls this weighted die and wins $1 if the outcome is in event , wins $ 2 if the outcome is in event , and loses $3 if the outcome is in event .

The expected value of this game is 0.70. The player should expect to win 0.70.

To find out the expected value of this game, we will use the formula

E(X)= ∑(xP(x))

where X represents the possible outcomes and P(x) represents the probabilities of those outcomes.

Let's determine the probabilities of each event:

Event 1: P(1) = 0.3

Event 2: P(2) = 0.5

Event 3: P(3) = 0.2

Using the information given in the question, we know that a player wins 1 if the outcome is in event 1, wins 2 if the outcome is in event 2, and loses $3 if the outcome is in event 3.

Therefore, the values of each event are:

Event 1: 1

Event 2: 2

Event 3: -3

Now, we can use the formula to calculate the expected value:

E(X)= ∑(xP(x))E(X)

      = (1 x 0.3) + (2 x 0.5) + (-3 x 0.2)

E(X) = 0.3 + 1 - 0.6E(X)

       = 0.7

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The area of the shape is 331km²

The area of shape is the space enclosed within the perimeter or the boundary of a given shape. Area is measured in units².

The shape above can be divided into 4 shapes.

The area of the first triangle = 1/2 bh

= 1/2 × 14 × 21

= 7× 21

= 147km²

The area of the first rectangle = l× b

= 14 × 5

= 70km²

The area of the second triangle = 1/2 bh

= 1/2 × 10× 18

= 10× 9

= 90km²

area of the second rectangle = l × b

= 4 × 6

= 24 km²

area of the shape = 147+90+70+24 = 331km²

Therefore the area of the shape is 331km²

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In online and blended learning environments, discussion-based assessments are often used as a way for students to engage in collaborative and reflective learning.

These assessments require students to participate in online discussions, often in small groups or with the whole class, to share ideas, ask questions, and build upon each other's thinking.

There are several ways that discussion-based assessments can be used in online and blended learning environments:

As a form of formative assessment:

Discussion-based assessments can be used as a way for teachers to check for student understanding and to provide feedback in real-time.

This can help teachers adjust their instruction to better meet the needs of their students.

As a way to encourage student engagement: Discussion-based assessments can help students feel more connected to their peers and to the learning community.

By providing opportunities for students to interact with each other, teachers can help students feel more invested in their learning.

As a way to promote critical thinking:

Discussion-based assessments can help students develop critical thinking skills by providing opportunities for them to analyze, evaluate, and synthesize information.

Through online discussions, students can learn how to articulate their own ideas and respond to the ideas of others.

As a way to promote collaborative learning:

Discussion-based assessments can help students develop collaborative learning skills by providing opportunities for them to work together to solve problems, share information, and build on each other's ideas.

To effectively use discussion-based assessments, teachers need to design questions and prompts that are thought-provoking and require students to engage in higher-order thinking.

Teachers also need to establish clear expectations for participation and provide guidelines for respectful and constructive discourse.

Finally, teachers should be actively engaged in the discussion and provide feedback and guidance to help students develop their thinking and communication skills.

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Answer:

|a+5|=a+5

Step-by-step explanation:

if a>-5 or a= -5 |a+5|=a+5

if a< -5 |a+5|= -a-5

Ian's score as a percentage is 73.3%, rounded to one decimal place. This means that he obtained 73.3% of the total marks possible in the test.

The percentages are a useful way to express scores or quantities relative to the whole. In this case, Ian's percentage score indicates how well he performed on the test compared to the maximum score possible.

To calculate Ian's score as a percentage, we can use the formula:

percentage score = (marks obtained / total marks) x 100

In this case, Ian scored 44 out of a total of 60 marks. Plugging these values into the formula, we get:

percentage score = (44 / 60) x 100 = 73.3%

Therefore, Ian's score as a percentage is 73.3%, rounded to one decimal place. This means that he obtained 73.3% of the total marks possible in the test.

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The expression 3z^2 - 27 can be simplified by factoring out the greatest common factor (GCF), which is 3:

3z^2 - 27 = 3(z^2 - 9)

Now, we can simplify the expression further by recognizing that the term inside the parentheses is the difference of squares, which can be factored as:

z^2 - 9 = (z + 3)(z - 3)

Substituting this back into our original expression, we get:

3z^2 - 27 = 3(z^2 - 9) = 3(z + 3)(z - 3)

Therefore, the simplified form of the expression 3z^2 - 27 is 3(z + 3)(z - 3).

The venn diagram is given below and A - (AnBnC) is {a, b, d}.

What is venn diagram?

A Venn diagram consists of one or more circles, each representing a set. The items that belong to each set are placed inside the appropriate circle, and the overlapping areas between the circles represent items that belong to both sets.

(a) Here's a Venn diagram that shows the relation of U, A, B, and C:

The circle labeled "A" represents the set A, the circle labeled "B" represents the set B, and the circle labeled "C" represents the set C. The regions where the circles overlap represent the elements that are in more than one set.

(b) A - (AnBnC) means the elements that are in set A but not in the intersection of sets A, B, and C. To find these elements, we first need to find the intersection of sets A, B, and C:

AnBnC = {f}

Then, we can subtract that set from set A:

A - (AnBnC) = {a, b, d}

Therefore, A - (AnBnC) is {a, b, d}.

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Answer:

1<4(3)+2

1<12+2

1<14

so therefore we say the answer is 1 is lesser than 14

Answer:

a)  The intercepts of the graph are (-75, 0), (75, 0), (0, -75) and (0, 75).

b)  The radius is 75 miles.

c)  The area of the region is 17,671 square miles (to the nearest square mile).

Step-by-step explanation:

The x-intercepts are the points at which the graph crosses the x-axis, so when y = 0. Therefore, to find the x-intercepts, substitute y = 0 into the given equation.

[tex]\implies x^2+y^2=5625[/tex]

[tex]\implies x^2+(0)^2&=5625[/tex]

[tex]\implies x^2&=5625[/tex]

[tex]\implies x&=\sqrt{5625}[/tex]

[tex]\implies x&=\pm75[/tex]

The y-intercepts are the points at which the graph crosses the y-axis, so when x = 0. Therefore, to find the y-intercepts, substitute x = 0 into the given equation.

[tex]\implies x^2+y^2=5625[/tex]

[tex]\implies (0)^2+y^2&=5625[/tex]

[tex]\implies y^2&=5625[/tex]

[tex]\implies y&=\sqrt{5625}[/tex]

[tex]\implies y&=\pm75[/tex]

Therefore, the intercepts of the graph are:

[tex]\hrulefill[/tex]

The general equation of a circle is:

[tex](x-h)^2+(y-k)^2=r^2[/tex]

where (h, k) is the center and r is the radius of the circle.

By comparing the given equation with the general equation:

Take the positive square root of r² to find the radius of the circle (since length cannot be negative):

[tex]\implies r=\sqrt{5625}[/tex]

[tex]\implies r=75[/tex]

Therefore, the radius of the circle is 75 miles.

[tex]\hrulefill[/tex]

The formula for the area of a circle is:

[tex]A=\pi r^2[/tex]

where r is the radius.

To find the area of the region in which the broadcast from the station can be picked up, find the area of the circle with the radius from part b, r = 75:

[tex]\implies A= \pi \cdot 75^2[/tex]

[tex]\implies A=5625 \pi[/tex]

[tex]\implies A=17671.4586...[/tex]

Therefore, the area of the region is 17,671 square miles (to the nearest square mile).

Answer:

Step-by-step explanation:

To find:-

Answer:-

The given equation of the circle is ,

[tex]:\sf\implies x^2 + y^2 = 5625 \\[/tex]

[tex]\rule{200}2[/tex]

G R A P H : -

[tex]\setlength{\unitlength}{7mm}\begin{picture}(0,0)\thicklines\qbezier(2.3,0)(2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,-2.121)(0,-2.3)\qbezier(2.3,0)(2.121,-2.121)(-0,-2.3)\put(0,0){\vector(1,0){6}}\put(0,0){\vector(-1,0){6}}\put(0,0){\vector(0,1){6}}\put(0,0){\vector(0,-1){6}}\put(.2,2.5){$\sf (0,75)$}\put(.2, - 2.7){$\sf (0,-75)$}\put(2.5,0.2){$\sf (75,0)$}\put( - 4.2,.2){$\sf ( - 75,0)$}\put(4,4){$\bigstar \: \: \sf Centre = (0,0) $}\put(4,3){$\bigstar \: \: \sf Radius = 75\ miles $}\put(4,-4){$\boxed{\bf \textcopyright \: \: Tony Stark }$}\end{picture}[/tex]

[tex]\rule{200}2[/tex]

Answer a :-

The intercepts are the points at which the circle cuts the x-axis and y-axis . At x intercept , the value of y coordinate becomes 0 and at y intercept the x coordinate becomes 0 .

So for finding x intercept , plug in y = 0 , in the given equation of circle, as ;

[tex]:\sf\implies x^2 + (0)^2 = 5625 \\[/tex]

[tex]:\sf\implies x^2 = 5625 \\[/tex]

[tex]:\sf\implies x =\sqrt{5625} \\[/tex]

[tex]:\sf\implies x =\pm 75 \\[/tex]

[tex]:\sf\implies \red{ x = +75 , -75} \\[/tex]

Hence the circle cuts the x-axis at (75,0) and (-75,0).

To find out y intercept plug in x = 0 in the given equation of the circle as ,

[tex]:\sf\implies (0)^2 + y^2 = 5625 \\[/tex]

[tex]:\sf\implies y^2 = 5625 \\[/tex]

[tex]:\sf\implies y=\sqrt{5626}\\[/tex]

[tex]:\sf\implies y = \pm 75 \\[/tex]

[tex]:\sf\implies \red{ y = +75,-75} \\[/tex]

Hence the circle cuts the y-axis at (0,75) and (0,-75).

[tex]\rule{200}2[/tex]

Answer b :-

Next we are interested in finding out the radius of the circle, for that we need to compare the given equation of circle to the standard equation of circle .

Standard equation of circle :-

[tex]:\sf\implies \red{ (x-h)^2 + (y-k)^2 = r^2}[/tex]

where ,

We can rewrite the given equation of circle as ,

[tex]:\sf\implies (x-0)^2 + (y-0)^2 = 5625 \\[/tex]

[tex]:\sf\implies (x-0)^2+(y-0)^2 = 75^2\\[/tex]

On comparing to the standard form, we have;

[tex]:\sf\implies \red{ radius = 75\ miles } \\[/tex]

Hence the radius of the circle is 75miles .

[tex]\rule{200}2[/tex]

Answer c :-

To find out the area we can use the formula of area of circle , which is ;

[tex]:\sf\implies \red{Area=\pi(radius)^2} \\[/tex]

We already got the radius to be 75 miles in the previous part of the question. Plugging in that value would give us the area , as ;

[tex]:\sf\implies Area =\pi (75\ miles )^2 \\[/tex]

[tex]:\sf\implies Area = \dfrac{22}{7}\times 5625\ miles^2\\[/tex]

[tex]:\sf\implies \red{ Area = 17678.57 \ miles^2 } \\[/tex]

Hence the area of the circle is 17678.57 miles² .

The radius of the cylinder is 4.5 cm.

What is a cylinder?

A cylinder is a three-dimensional solid in mathematics that maintains two parallel bases that are separated at a fixed distance by a curving surface. An axis joins the centres of each base, and the bases are often circular in shape.

We are given that the sum of the height and radius of a right circular cylinder is 9 cm.

This means that r + h = 9

We are also given that the total surface area is 81π cm square.

So,

⇒ 81π = 2πr (r + h)

⇒ 81 = 2 * r * 9

⇒ 81 = 18r

⇒ r = 4.5 cm

Hence, the radius of the cylinder is 4.5 cm.

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Answer:

$35.35

Step-by-step explanation:

First, let's add all these prices together:

$8.95 + $11.49 + $12.75 = $33.19

Then, let's find how much will the whole price increase with the taxes:

[tex] \frac{33.19 \times 6.5\%}{100\%} = 2.15735[/tex]

And finally, let's add this number and the old price together:

$33.19 + $2.15735 = $35.34735 ≈ $35.35

x = 18 is given by:

f(x) = (x - 18)²
      -------------

             x

Or, if you have choices it is CHOICE A :) :P

Answer:

I can see the pattern. There will be 30 toothpicks to make 10 squares accordingto your pattern. The pattern with an expression is +3.

Step-by-step explanation:

If you list the 4,7,10 and see the pattern between these numbers. You are adding 3 to the 4 to the 7 and 7 to the 10. Then, follow the pattern until you get 10 squares.

The numbers in scientific notation 3.26E21 and 5.95E-27 are 3.26E21 = 3.26 * 10^21 and 5.95E-27 = 5.95 * 10^-27

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

3.26E21 and 5.95E-27

A number represented using aEb can be represented as

aEb = a * 10^b

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

3.26E21 = 3.26 * 10^21

5.95E-27 = 5.95 * 10^-27

Hence, the numbers in scientific notation are 3.26E21 = 3.26 * 10^21 and 5.95E-27 = 5.95 * 10^-27

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A city has a population of 28,000 people. Suppose that each year the population grows by 4.5%. What will the population be in 13 years?

Answer: Approximately 496,21 people

This is an exponential growth question, specifically population growth. We will use the following equation to help us solve this situation:

[tex]P(t) = a(1+r)^t[/tex]

[tex]P(t) = \text{total population} \ (??)[/tex]

[tex]\text{a = initial population = 28,000}[/tex]

[tex]\text{r = growth factor (the percent as a decimal) = 0.045}[/tex]

[tex]1 + r = 1 + 0.045 = 1.045[/tex]

[tex]\text{t = time in years = 13}[/tex]

So, we can substitute all our information and solve:

[tex]P(13) = 28,000(1.045)^{13} = 496,21.49[/tex]

Rounding to the nearest whole person, 496,21 people in 13 years.

The probability that the first person who walks in wearing an N95 mask is the third person is 0.128 or 12.8%. The probability that the first person who walks in wearing an N95 mask is the third person can be found using the geometric probability distribution formula.

The formula for the probability of the first success occurring on the kth trial is:

P(k) = (1 - p)^(k-1) * p

Where p is the probability of success (in this case, 20% or 0.2), and k is the number of trials.

In this case, we want to find the probability that the third person who walks in is wearing an N95 mask, so k = 3.

Substituting the values, we get:

P(3) = (1 - 0.2)^(3-1) * 0.2

P(3) = 0.64 * 0.2

P(3) = 0.128

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Answer:

-2299

Step-by-step explanation:

12(-72) = -864

(-3 - 32) = -35

41(-35) = -1435

-864 + -1435 = -2299

Number of students should be sampled to be within 0.5 drink of population mean with 95% probability is 617 students.

To determine the sample size required to estimate the population mean with a given level of precision, we can use the formula for the margin of error

Margin of error = Z × (standard deviation / sqrt(sample size))

where Z is the critical value of the standard normal distribution corresponding to the desired level of confidence. For a 95% confidence level, Z is 1.96.

We want the margin of error to be no more than 0.5 drinks, so we can set up the equation

0.5 = 1.96 × (6.3 / sqrt(sample size))

Solving for the sample size, we get

sqrt(sample size) = 1.96 × 6.3 / 0.5

sqrt(sample size) = 24.82

sample size = (24.82)^2

sample size = 617

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If Pancake Paradise makes the best Pineapple Pancakes. Each batch makes 3 whole pancakes.

To find the number of whole pancakes in each batch, we need to first calculate the area of one pancake.

The diameter of each pancake is 2.5 inches, which means the radius is 1.25 inches (diameter = 2 x radius). The area of each pancake is:

A = πr^2 = π(1.25 in)^2 ≈ 4.91 in^2

Next, we need to determine how many pancakes can be made from a batch of batter that covers an area of 15 inches.

Number of pancakes = Total area / Area of one pancake

Number of pancakes = 15 in^2 / 4.91 in^2 ≈ 3.05

Since we can't make a fraction of a pancake, we need to round down to the nearest whole number. Therefore, each batch makes 3 whole pancakes.

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a.  At the moment when the batter is halfway to first base, his distance from second base is not changing.

b. His distance from third base increasing at the same moment at rate 6√2 ft/s.

Let A be the starting point of the batter, B be second base and C be third base, and $P$ be the position of the batter when he is halfway to first base.

Note that AP = AB/2 = 45 ft, B = AB - AP = 45 ft, and PC = AC - AP = 45√2 ft.

(a) We want to find [tex]$\frac{d}{dt} PB$[/tex]  when AP = 45 ft and [tex]$\frac{dAP}{dt} = 24$[/tex] ft/s.

By the Pythagorean theorem, we have

PB² = AB² - AP² = 90² - 45² = 4050, so PB = √4050 ft.

To find [tex]$\frac{dPB}{dt}$[/tex] , we take the derivative of both sides of the equation PB² = 4050 with respect to time t:

[tex]2PB\frac{dPB}{dt} &= 0 \\\\\\\frac{dPB}{dt} &= 0[/tex]

Therefore, at the moment when the batter is halfway to first base, his distance from second base is not changing.

(b) We want to find [tex]$\frac{d}{dt} PC$[/tex] when AP = 45 ft and [tex]$\frac{dAP}{dt}[/tex] = 24 ft/s.

By the Pythagorean theorem, we have

PC² = AC² - AP² = (90\√2})² - 45² = 5850,

so PC = √5850 ft.

To find [tex]$\frac{dPC}{dt}$[/tex] we take the derivative of both sides of the equation PC² = 5850 with respect to time t:

[tex]2PC\frac{dPC}{dt} &= 2\frac{dAP}{dt}(AC - AP) \\\\\frac{dPC}{dt} &= \frac{dAP}{dt} \frac{AC - AP}{PC} \&= \frac{24(90\sqrt{2} - 45)}{\sqrt{5850}} \&= 16\sqrt{2} \approx 22.63 \text{ ft/s}.\end{align*}[/tex]

Therefore, at the moment when the batter is halfway to first base, his distance from third base is increasing at a rate of 16√2 ft/s.

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The expected recorded score of the given data of short quiz is,

Expected recorded score is 8.5.

And maximum of the two expected recorded score is 12.45.

Expected recorded score e(x + y),

e(x + y) = ΣΣ (x + y)  p(x, y)

Simplify this expression by rearranging the sum,

e(x + y) = ΣΣ x p(x, y) + ΣΣ y  p(x, y)

Using the table given,

ΣΣ x p(x, y)

= (0×0.02) + (5×0.04) + (10×0.01) + (0×0.06) + (5×0.13) + (15×0.15) + (0×0.02) + (5×0.20) + (10×0.16)

= 4.6

ΣΣ y p(x, y)

= (0×0.02) + (0×0.06) + (0×0.02) + (5×0.10) + (5×0.13) + (5×0.01) + (10×0.20) + (10×0.15) + (10×0.16)

= 3.9

Expected recorded score is,

e(x + y)

= 4.6 + 3.9

= 8.5

The maximum of the two scores is recorded,

Then the recorded score will be either x or y, whichever is larger.

Probability distribution of the maximum score,

P(max(x, y) = 0)

= P(x=0, y=0)

= 0.02

P(max(x, y) = 5)

= P(x=5, y=0) + P(x=0, y=5) + P(x=5, y=5)

= 0.06 + 0.04 + 0.13

= 0.23

P(max(x ,y) = 10)

= P(x=10, y=0) + P(x=0, y=10) + P(x=10, y=5) + P(x=5, y=10) + P(x=10, y=10)

= 0.01 + 0.15 + 0.20 + 0.10 + 0.16

= 0.62

P(max(x, y) = 15)

= P(x=15, y=0) + P(x=0, y=15) + P(x=15, y=5) + P(x=5, y=15) + P(x=15, y=10) + P(x=10, y=15) + P(x=15, y=15)

= 0.01 + 0.10 + 0.10 + 0.10 + 0.01 + 0.01 + 0.01

= 0.34

Expected recorded score can be calculated as the weighted sum of the possible maximum scores,

E(max(x, y)) = Σ max(x, y) × P(max(x, y))

Substituting the probabilities, we get,

E(max(x ,y))

= 0×0.02 + 5×0.23 + 10×0.62 + 15×0.34

= 0+ 1.15 + 6.2 + 5.1

= 12.45

Expected recorded score is 12.45 (upto two decimal places).

Therefore, expected recorded score is 8.5 and maximum of the expected recorded score is 12.45.

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The above question is incomplete, the complete question is:

An instructor has given a short quiz consisting of two parts. for a randomly selected student, let x = the number of points earned on the first part and y = the number of points earned on the second part. suppose that the joint pmf of x and y is given in the accompanying table.

                                                 y

     p(x, y)                     0               5                  10                               15

    0                           0.02          0.06              0.02                          0.10

x   5                           0.04           0.13              0.20                          0.10

   10                           0.01            0.15              0.16                          0.01

(a) if the score recorded in the grade book is the total number of points earned on the two parts, what is the expected recorded score e(x + y)? (enter your answer to one decimal place.)

(b) if the maximum of the two scores is recorded, what is the expected recorded score? (enter your answer to two decimal places.)