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Lesson 02. 05: Module Two Project-Based Assessment


Printable Assessment Module Two Project Based Assessment


Module Two Project-Based Assessment


Part 1


The table shows the measurements of shooting stars that were measured. Use the table


to complete the activities below.


Shooting star length


(in feet)


Number


10


2


8


10


6


8


6


10


7


8


10


금


4


1. Compare the sizes. Think about the number of Xs that would appear on the line plot.


Write the shooting star lengths in the correct box.


Fewer than 5 Xs


More than 5 Xs


COM


10


2. Complete the line plot for the given set of data.


Lengths of Shooting Stars


7


O


2


Measurement in feet

The 9th girl ate 51 candies.

Let the number of candies eaten by the 9th girl be x.

The average number of candies eaten by 8 girls is given as (210/x+210)/8, which simplifies to 210/8 + x/8.

After the 9th girl eats x candies, the total number of candies eaten becomes 210 + x.

The new average is given as (210 + x)/9 = 29.

Solving for x, we get:

210 + x = 261

x = 51

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Note that the expected value of the retailers profit is - $114. This means he made a loss.

To find the expected value we must proceed as follows

Expected Value - E(F) is

Probability of F - P(F)

= 100 x ($100 - $200) + (P(F) = $50) x ($50 - $200) + (P(F) = $ -200) x ( $ - 200 - $200) +  (P(F)   = $- 350) x ($ -350  $ 200)

= (0.9) x (-100) + (0.07 ) x (-150) + (0.01)  x  (-550)  + (0.02) x (-400)

= - 90  - 10.5  - 5.5 -8

E(F) = $ -114

So it is right to state that the expected value of the retailer's profit per protection plan sold is -$114, which  is a loss.

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Full Question:

An electronics retailer sells mobile phones with an optional protection plan for $100. In case of an accident, the customer pays a $50 deductible and the retailer covers the rest of the repair cost, which is typically $200 per repair. The protection plan covers a maximum of 333 repairs.

Let X be the number of repairs a randomly chosen customer uses under the protection plan, and let F be the retailer's profit from one of these protection plans. The probability distributions of X and F are:

X = number of repairs: 0 1 2 3

Probability: 0.90 0.07 0.02 0.01

F = retailer profit: $100-$50-$200-$350

Probability: 0.90 0.07 0.02 0.01

The task is to find the expected value of the retailer's profit per protection plan sold.

If 13 matches are played in total then, 7 2-player matches and 6 6-player matches are played.

Here we see that the table has two columns- 6 player Athletes and 2 player athletes. It is given that no athlete participates in both the type of games. Hence we can say that

If one match for 2 player game is held then 2 players are employed there.

Hence we have 48 players left

hence we will have 48/6 = 8 6-player matches.

Similarly, if 1 6-player match is played then 44 players applied for the 2-player match, hence, we have 44/2 = 22 2-player matches

If 4 2-player matches are held then we will have 8 players booked. Hence 42/6 = 7 6-player matches were held.

If 4 6-player matches were held then, we have 26/2 = 13 2-player matches.

Hence the table will be

Number of 6 Player Games                       Number of 2-player games

                     8                                                                 1

                     1                                                                 22

                     7                                                                 4

                     4                                                                 13

b)

Let the total 2-player games played be x and 6-player games be y

we have,

x + y = 13

2x + 6y = 50

or, 2(x + y) + 4y = 50

or, 26 + 4y = 50

or, 4y = 24

or, y = 6

Hence x = 7

Therefore, in total 7 2-player matches and 6 6-player matches are played.

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The MAD for Doctor A's data set is 0.67 and the MAD for Doctor B's data set is 0.83. Doctor A's data set has less variation than Doctor B's data set, as indicated by their respective MADs.

To calculate the mean absolute deviation (MAD) for a data set, we first find the mean of the data set, and then find the absolute deviation of each value from the mean. We then find the mean of these absolute deviations.

a) For Doctor A's data set on corrective lenses, the mean is:

Mean = (15+18+17+16+14)/5 = 16

The absolute deviations from the mean are:

|15-16| = 1

|18-16| = 2

|17-16| = 1

|16-16| = 0

|14-16| = 2

The mean of these absolute deviations is:

MAD = (1+2+1+0+2)/5 = 1.2

Therefore, the MAD for Doctor A's data set is 1.2.

b) For Doctor B's data set on corrective lenses, the mean is:

Mean = (17+19+20+16+18)/5 = 18

The absolute deviations from the mean are:

|17-18| = 1

|19-18| = 1

|20-18| = 2

|16-18| = 2

|18-18| = 0

The mean of these absolute deviations is:

MAD = (1+1+2+2+0)/5 = 1.2

Therefore, the MAD for Doctor B's data set is also 1.2.

Comparing the two data sets using their MAD, we can see that they have the same amount of variation or dispersion from the mean. Both sets have a MAD of 1.2, indicating that the average absolute deviation of each value from the mean is the same for both sets.

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Answer:

Doctor A MAD: 11.8

Doctor B MAD: 9.32

Step-by-step explanation:

This is what I got on the assignment.

a) The probability that, for at least 15 packs, the average weight of the erasers in the pack is at least 31.95 g is approximately 0.0384.

b) The probability that, on average, the unit finds at least 37.2 "good" erasers per day is approximately 0.3133.

a) To solve this problem, we need to use the central limit theorem. According to this theorem, the distribution of sample means becomes approximately normal, regardless of the shape of the population distribution, when the sample size is sufficiently large (usually, n >= 30). In this case, since the sample size is 45, we can assume that the distribution of sample means will be approximately normal.

Now, we need to find the probability that the average weight of at least 15 packs is at least 31.95 g. We can use the normal distribution to calculate this probability. We first calculate the z-score for this value as follows:

z = (31.95 - 31.9) / (0.163 / √(45)) = 1.77

Using a standard normal table or calculator, we can find the probability that a z-score is greater than or equal to 1.77. This probability is approximately 0.0384.

b) To solve this problem, we need to use the normal approximation to the binomial distribution. Since each eraser is either "good" or "bad", the number of "good" erasers that the unit finds each day follows a binomial distribution with parameters n = 50 and p = probability of finding a "good" eraser = (32.3 - 31.7)/(32.3 - 31.5) = 0.5.

Now, we need to find the probability that, on average, the unit finds at least 37.2 "good" erasers per day. We can use the normal distribution to calculate this probability. We first calculate the z-score for this value as follows:

z = (37.2 - 25) / 25 = 0.488

Using a standard normal table or calculator, we can find the probability that a z-score is greater than or equal to 0.488. This probability is approximately 0.3133.

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Answer:

The value of 1/4 (-4^3+10^2) is 9

Div(f) = 2cos(x) + [tex]5e^y[/tex], and curl(f) = < 0, 9sin(x), 5e^y >.

To calculate div(f) and curl(f), we need to express f as a vector field:

f = < 2 sin(x), [tex]5e^y[/tex][tex]5e^y[/tex], 9 cos(x) >

Then, we can use the formulas for divergence and curl:

div(f) = ∂f₁/∂x + ∂f₂/∂y + ∂f₃/∂z

curl(f) = < ∂f₃/∂y - ∂f₂/∂z, ∂f₁/∂z - ∂f₃/∂x, ∂f₂/∂x - ∂f₁/∂y >

Let's compute these step by step:

div(f) = ∂f₁/∂x + ∂f₂/∂y + ∂f₃/∂z

= 2cos(x) + [tex]5e^y[/tex] + 0

= 2cos(x) + [tex]5e^y[/tex]

curl(f) = < ∂f₃/∂y - ∂f₂/∂z, ∂f₁/∂z - ∂f₃/∂x, ∂f₂/∂x - ∂f₁/∂y >

= < 0 - 0, 0 - (-9sin(x)), 5e^y - 0 >

= < 0, 9sin(x), 5e^y >

Therefore, div(f) = [tex]2cos(x) + 5e^y[/tex], and curl(f) = [tex]< 0, 9sin(x), 5e^y > .[/tex]

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If x > 2.5, both expressions within absolute value symbols are positive.

The equation becomes: y = (2x + 5) - (2x - 5) = 10.

For the given intervals of x:

If x < -2.5, both expressions within absolute value symbols are negative. Thus, the equation is: y = -(2x + 5) - (-(2x - 5)) = -10.

If x > 2.5, both expressions within absolute value symbols are positive.

The equation becomes: y = (2x + 5) - (2x - 5) = 10.

If -2.5 ≤ x ≤ 2.5, the first expression is positive and the second is negative.

The equation is: y = (2x + 5) - (-(2x - 5)) = 4x.

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1. All square, upper triangular matrices are diagonalizable. (TRUE)

2. If a matrix is diagonalizable, then it is invertible. (FALSE)

1. True: All square, upper triangular matrices are diagonalizable.

A square matrix is diagonalizable if it can be transformed into a diagonal matrix by a similarity transformation, which means there exists an invertible matrix P such that P^-1 * A * P is a diagonal matrix. Upper triangular matrices have all their elements below the main diagonal equal to zero.

Since the eigenvalues of an upper triangular matrix are equal to its diagonal elements, we can form a diagonal matrix with these eigenvalues. Since there exists such a diagonal matrix, all square, upper triangular matrices are diagonalizable.

2. False: If a matrix is diagonalizable, it is not necessarily invertible

Diagonalizable matrices can be transformed into a diagonal matrix with the eigenvalues along the main diagonal.

However, invertibility requires that the matrix have a nonzero determinant, which means that all of its eigenvalues must be nonzero.

If a diagonalizable matrix has a zero eigenvalue, its determinant will be zero, and it will not be invertible. Therefore, diagonalizability does not guarantee invertibility.

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Answer:

N'(3, 2), M'(0, 1), O'(1, 3)

Step-by-step explanation:

Please sketch the graphs of the two triangles and the line of reflection to confirm my answer.

Since NMO is reflected over the line

x = -1, the y-coordinates of N'M'O' will remain the same. Since the x-coordinate of N is at -5, which is 4 units to the left of the line of reflection, the x-coordinate of N' is at -1 + 4 = 3. Since the x-coordinate of M is at -2, which is 1 unit to the left of from the line of reflection, the x-coordinate of M' is at -1 + 1 = 0. Since the x-coordinate of O is at -3, which is 2 units to the left of the line of reflection, the x-coordinate of O' is at -1 + 2 = 1. So the vertices of N'M'O' are at (3, 2), (0, 1), and (1, 3).

The role of the brackets, and how they effects the sum is given as the inside signs get changed after the opening of the bracket.

The associative property of addition is a mathematical statement that asserts that the arrangement of three or more integers does not affect their total. This indicates that no matter how the numbers are organised, the total of three or more integers remains the same.

The associative property of addition is a mathematical principle that asserts that when adding three or more integers, the amount obtained is constant regardless of how the numbers are grouped. Grouping here refers to where the brackets are positioned.

The sum for the 1st term is 10 + 7 – 5 + 3 = 15

The sum of the 2nd term is 10 + 7 – (5 + 3) = 17 - 8 = 9

Here, the bracket used made all the difference so after opening of the bracket the sign inside changed which impacted the summation of the terms above.

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Answer:

approximately 42.85 of whatever unit

The windshield sector area is 706.86 cm².

To calculate the sector area of the windshield wiper, we need to use the formula for the area of a sector of a circle. The formula is:

A = (θ/360°) x πr²

where A is the area of the sector, θ is the central angle of the sector in degrees, and r is the radius of the circle.

In this problem, we are given that the windshield wiper has a length of 45 cm, which means that the radius of the circle traced by the wiper is 45 cm/2 = 22.5 cm.

We are also given that the wiper creates a central angle of 120° in one wipe. Substituting these values into the formula, we get:

A = (120°/360°) x π(22.5 cm)²

A = (1/3) x π x (22.5 cm)²

A ≈ 706.86 cm²

Therefore, the sector area of the windshield wiper is approximately 706.86 square centimeters.

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According to the information, the customer would save $492 in the first year by switching to Intellivision; the customer would save $207 in the second year by using Intellivision; ElectroniSource would be cheaper in the third year.

a. To calculate the annual cost for ElectroniSource: $42/month for phone service x 12 months = $504/year, $35/month for internet service x 12 months = $420/year, and $59/month for cable television x 12 months = $708/year.

So the total annual cost with ElectroniSource would be $504 + $420 + $708 = $1,632.

With Intellivision, the flat monthly fee for all three services is $95, so the total annual cost would be $95 x 12 months = $1,140.

Therefore, the customer would save $1,632 - $1,140 = $492 in the first year by switching to Intellivision.

b. After the first year, Intellivision raises the rates by 25%, so the new monthly fee would be $95 x 1.25 = $118.75.

The total annual cost in the second year would be $118.75 x 12 months = $1,425.

Using the same services, the annual cost with ElectroniSource would still be $1,632.

Therefore, the customer would save $1,632 - $1,425 = $207 in the second year by using Intellivision.

c. If Intellivision raises the rates by 16% in the third year compared to the second year, the new monthly fee would be $118.75 x 1.16 = $137.95.

The total annual cost in the third year would be $137.95 x 12 months = $1,655.4.

The annual cost with ElectroniSource would still be $1,632.

Therefore, ElectroniSource would be cheaper in the third year.

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The quotient of the division 10x² + 3x - 77 ÷ 2x + 7 is 5x - 16

The quotient expression is given as

10x² + 3x - 77 ÷ 2x + 7

The long division expression is represented as

2x + 7 | 10x² + 3x - 77

So, we have the following division process

            5x - 16

2x + 7 | 10x² + 3x - 77

            10x²  + 35x

          --------------------------------

             -32x - 77

             -32x - 112

          -------------------------------------

                      35

Hence, the quotient of the long division is 5x - 16

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You should always measure your following distance in seconds. The correct answer for measuring following distance is A.  

It is important to keep a safe distance between vehicles on the road, which is known as following distance. Measuring following distance in seconds is a more reliable method than measuring it in car lengths or feet.

The recommended following distance is typically two to three seconds. This means that when the vehicle in front of you passes a fixed point on the road, you should not reach that same point before two to three seconds have elapsed.

Measuring in seconds is more effective as it takes into account the speed at which both vehicles are traveling. If you measure in car lengths or feet, it may not be accurate as the length of the cars may differ, and it does not account for the speed of the vehicles. By measuring in seconds, you can maintain a safe distance and reduce the risk of accidents or collisions.

The correct answer for measuring following distance is A.  

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Complete question is:

You should always measure your following distance in __________

A. seconds. B. car lengths. C. feet.

The number of cakes that can be made with a large bag of sugar, we first need to determine the amount of sugar in a small bag and then calculate the amount of sugar needed for one cake.

1. Find the amount of sugar in a small bag:
Since the large bag contains 30 cups of sugar and is 2.5 times larger than the small bag, we can write the equation:
Small bag = Large bag / 2.5
Small bag = 30 cups / 2.5
Small bag = 12 cups of sugar

2. Determine the amount of sugar needed for one cake:
The small bag contains enough sugar to make 9 cakes and have 0.75 cups of sugar remaining. So, we can subtract the remaining sugar from the total amount in the small bag:
Sugar used for 9 cakes = 12 cups - 0.75 cups
Sugar used for 9 cakes = 11.25 cups

Now, we can find the amount of sugar needed for one cake:
Sugar per cake = Sugar used for 9 cakes / 9
Sugar per cake = 11.25 cups / 9
Sugar per cake = 1.25 cups

3. Calculate the number of cakes that can be made with a large bag of sugar:
Cakes from large bag = Large bag sugar / Sugar per cake
Cakes from large bag = 30 cups / 1.25 cups
Cakes from large bag = 24

Therefore, a baker can make 24 cakes with a large bag of sugar.

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The number of calories in the 3 ounce bar is 480.

To solve this problem, we can use a proportion. If the 2-ounce protein bar has 320 calories, then we can set up the following proportion:

2 oz / 320 cal = 3 oz / x cal

where x is the number of calories in the 3-ounce protein bar.

We can solve for x by cross-multiplying and simplifying:

2 oz * x cal = 3 oz * 320 cal

2x = 960x = 480

Therefore, the 3-ounce protein bar has 480 calories.

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Answer: hopefully the image helps

Step-by-step explanation:

The mean is equal to the median, so the data is symmetrical.

Given that :

Colin surveyed 12 teachers at his school to determine how much each person budgets for lunch.

The data is :

10   5   8   10   12   6   8   10   15   6   12   18

Mean is the average of these numbers.

Mean = (10 + 5 + 8 + 10 + 12 + 6 + 8 + 10 + 15 + 6 + 12 + 18) / 12

         = 10

Now median is the element in the middle arranged in an order.

Arrange the data in ascending order.

5  6  6  8  8  10  10  10  12  12  15  18

The middle element is the average of the 6th and 7th elements.

Median = (10  10) / 2 = 10

So mean and the median is equal. So the data is symmetrical.

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To integrate the function ∫(x² - 36) dx, we first need to factor out the expression inside the parentheses:
∫(x² - 36) dx = ∫(x - 6)(x + 6) dx

We can then use the power rule of integration to find the antiderivative:
∫(x - 6)(x + 6) dx = (1/3)x³ - 6x + C, where C is the constant of integration.

Since the original problem states X > 6, we can evaluate the definite integral using these limits:
∫(x² - 36) dx from 6 to X = [(1/3)X³ - 6X] - [(1/3)(6)³ - 6(6)]
= (1/3)X³ - 6X - 68

Therefore, the answer in exact form is (1/3)X³ - 6X - 68.

To integrate the given function, first note the correct notation for the function: ∫(x^2)/(x^2 - 36) dx for x > 6.

To solve this, we can use partial fraction decomposition. The given function can be rewritten as:

∫(A(x - 6) + B(x + 6))/(x^2 - 36) dx

Solving for A and B, we find that A = 1/12 and B = -1/12. Now we rewrite the integral as:

∫[(1/12)(x - 6) - (1/12)(x + 6)]/(x^2 - 36) dx

Next, separate the two terms and integrate them individually:

(1/12)∫[(x - 6)]/(x^2 - 36) dx - (1/12)∫[(x + 6)]/(x^2 - 36) dx

Now, notice that the integrals are of the form ∫u'/u dx. The integral of this form is ln|u|. So we have:

(1/12)[ln|(x - 6)| - ln|(x + 6)|] + C

Using the logarithm property, we can rewrite the answer as:

(1/12)ln|((x - 6)/(x + 6))| + C

That is the exact form of the antiderivative for the given function.

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If in Angle STU, the measure of U=90°, the measure of S=31°, and TU = 77 feet, then the length of US to the nearest tenth of a foot is approximately 39.4 feet.

In angle STU, we have a right triangle with U=90°, S=31°, and TU=77 feet. To find the length of US, we can use the sine function:

sin(S) = opposite side (US) / hypotenuse (TU)

sin(31°) = US / 77 feet

To find the length of US, multiply both sides by 77 feet:

US = 77 feet * sin(31°)

US ≈ 39.4 feet

Therefore, the length of US to the nearest tenth of a foot is approximately 39.4 feet.

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To calculate the volume of the moon, we can use the formula V = (4/3)πr^3, where r is the radius of the moon.

Given that the radius of the moon is about 1.738 mega meters (or 1,738,000 meters), we can substitute this value into the formula and simplify as follows: V = (4/3) × 3.14 × (1.738 × 10^6)^3 V ≈ 2.196 × 10^19 cubic meters Therefore, approximately, the volume of the moon is 2.196 × 10^19 cubic meters.

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The value of 'a' is 2/3.

The problem presents a ratio, 4:15, that is equal to a ratio involving 'a' and 2 1/2. To solve for 'a', we need to isolate it on one side of the equation by cross-multiplying.

In the first step, we convert 2 1/2 to an improper fraction, 5/2, so that we can use it in the equation. We then cross-multiply by multiplying both sides of the equation by 5/2.

This eliminates the denominator on the right-hand side and simplifies the left-hand side.

Solve for 'a'

To solve for 'a', we can use cross-multiplication.

First, we need to convert 2 1/2 to an improper fraction:

2 1/2 = 5/2

Now we can write the equation as:

4/15 = a/(5/2)

To solve for 'a', we cross-multiply:

4/15 * 5/2 = a

a = 2/3

Finally, we solve for 'a' by multiplying 4/15 by 5/2 and simplifying the result. The answer is 2/3, which represents the value of 'a'.

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Answer is 27,529.82
To calculate the total purchase price of the car including tax, license, and title, we need to add the down payment, the financed balance, the sales tax, and the license and title charges.

First, we calculate the down payment:
10% of $25,625 = $2,562.50

Next, we calculate the financed balance:
$25,625 - $2,562.50 = $23,062.50

Then, we calculate the sales tax:
7.5% of $23,062.50 = $1,729.69

Finally, we add the license and title charges:
$1,729.69 + $175.13 = $1,904.82

So the total purchase price of the car including tax, license, and title is:
$2,562.50 + $23,062.50 + $1,904.82 = $27,529.82

Rounded to the nearest cent, the answer is option D: $27,735.14.

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Orlando should consider using simple interest and the amount he will have to pay is $696, under the condition that he wants to borrow $3,000 for the purchase of a used car. He needs to clear the loan after 4 years.

Orlando should apply the simple interest method.

The amount of interest he will pay using simple interest is evaluated

I = P × r × t

Here

I = interest paid

P = borrowed principal amount

r =  rate of annual interest

t = time

In this case,

P = $3,000

r = 5.8%

t = 4 years

Therefore,

I = $3,000 × 0.058 × 4

= $696

So Orlando will pay $696 in interest using simple interest.

The amount of interest he will pay using compound interest is calculated as follows:

[tex]A = P * (1 + r/n)^{(n*t)}[/tex]

I = A - P

Here,

A = end term amount

n = count of interest that is compounded each year

In this case,

P = $3,000

r = 5.2%

t = 4 years

Interest is compounded annually so n=1

Therefore,

A = $3,000 × (1 + 0.052/1)⁴

= $3,697.47

I = $3,697.47 - $3,000

= $697.47

So Orlando will pay $697.47 in interest using compound interest.

Therefore, Orlando should choose simple interest method and he will owe $1.47 less using that method.

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The following are correct about the triangle;

1. angle C is 60°

2. angle B is 60°

3. The length of segment DB is 3

4. The length of side x is 3√3

An equilateral triangle is a type of triangle in which all it's sides and angles are equal.

Since all the angles of an equilateral triangle are equal, then,

x+x+x = 180

3x = 180

x = 180/3 = 60°

therefore each angle is 60°

angle C and angle B are 60°

Using Pythagorean theorem

x² = 6²- 3²

x² = 36-9

x² = 27

x = √27

x = √9×3

x = 3√3

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Daniel will read approximately 193 pages in one hour if he continues reading at the same rate.

Since Daniel read 77 pages in $\frac{2}{5}$ of an hour, we can set up the proportion:

$\frac{77}{\frac{2}{5}} = x$

where $x$ is the number of pages he would read in 1 hour.

To solve for $x$, we can simplify the left side of the equation by multiplying both the numerator and denominator by 5, which gives:

$\frac{77 \times 5}{2} = x$

Simplifying this expression gives:

$192.5 = x$

Therefore, if Daniel continues reading at the same rate, he will read 192.5 pages in one hour. However, since the number of pages must be a whole number, we can round this answer to the nearest whole number to get:

$x \approx 193$

So, Daniel will read approximately 193 pages in one hour if he continues reading at the same rate.

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Answer:

Step-by-step explanation:

The correct answer is:

the initial height of the plant

The y-intercept represents the point where the graph intersects the y-axis. In the context of a relationship between plant growth and time, the y-intercept would represent the initial height of the plant, which is the height of the plant at the start of the experiment, before any growth has occurred.

When a number is chosen from set x, there is a 0.2 chance that the chosen number will also be found in sets y and z. The answer is option (B). 0.2.

The ratio of favourable outcomes to all possible outcomes of an event is known as the probability. The symbol x can be used to express the quantity of successful outcomes for a study with 'n' outcomes. The probability formula determines the likelihood that an event will occur.

It is the ratio of effective results to all effective results. The study of probability is a branch of mathematics that examines the likelihood that an event will occur. Probability, which expresses the likelihood that an event will occur, is calculated by dividing the total number of occurrences by the total number of positive events.

The number of elements of Both Y and Z answer is 5 and 25.

The probability of selecting a number from x that is an element of y and z

Intersection of y and z is (5, 25)

No.of elements in x is = 2/10

= 1/5

= 0.2

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