John recently scored 113 on a particular standardized achievement test. The scores on the test are distributed with a mean of 100 and a standard deviation of 10. His cousin, Brandon, took a different standardized test and scored 263. The scores on Brandon's test have a mean of 250 and a standard deviation of 25. Which student did relatively better on his particular test? A) John did better on his test. B) Brandon did better on his test. C) They both performed equally as well on their respective tests. D) It is impossible to tell since they did not take the same test. E) It is impossible to tell since the number of students taking the test is unknown.

Answer:Therefore, the solutions to the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 are x = 0, x = 1, x = 5.

Step-by-step explanation:

To solve the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0, we can use factoring by grouping.

First, we can group the first two and last two terms:

x^4 - 5x^3 + 7x^2 - 5x + 6 = (x^4 - 5x^3) + (7x^2 - 5x + 6)

Next, we can factor out x^3 from the first group and factor out 1 from the second group:

(x^3(x - 5)) + (7x^2 - 5x + 6)

Now, we can group the last two terms of the second group:

x^3(x - 5) + (7x^2 - 3x - 2x + 6)

Then, we can factor out 1 from the terms inside the parentheses and group them:

x^3(x - 5) + (7x^2 - 3x) + (-2x + 6)

Now, we can factor out x from the second and third groups:

x^3(x - 5) + x(7x - 3) - 2( x - 3)

We can simplify the third group by distributing the negative sign:

x^3(x - 5) + x(7x - 3) - 2x + 6

Finally, we can combine the second and third groups:

x^3(x - 5) + x(7x - 5) + 6

So, the factored form of the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 is:

(x^3(x - 5) + x(7x - 5) + 6) = 0

This equation can be solved by setting each factor equal to zero and solving for x:

x^3(x - 5) + x(7x - 5) + 6 = 0

(x^3 - 7x^2 + 5x) + (6 - 5x) = 0

x(x^2 - 7x + 5) - (5x - 6) = 0

x(x - 5)(x - 1) - (5x - 6) = 0

x(x - 5)(x - 1) = 5x - 6

x^3 - 6x^2 + 10x - 6 = 5x - 6

x^3 - 6x^2 + 5x = 0

x(x^2 - 6x + 5) = 0

x(x - 1)(x - 5) = 0

Therefore, the solutions to the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 are x = 0, x = 1, x = 5.

Answer:

[tex]-\cfrac{8x}{x^2-4}[/tex]

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

Simplify in below steps:

[tex]\cfrac{x-2}{x+2} -\cfrac{x+2}{x-2} =[/tex]                                   Bring to common denominator

[tex]\cfrac{(x-2)^2}{(x+2)(x-2)} -\cfrac{(x+2)^2}{(x+2)(x-2)}=[/tex]       Multiply both fractions by x +2 or x - 2

[tex]\cfrac{(x-2)^2-(x+2)^2}{(x+2)(x-2)}=[/tex]                           Make a single fraction

[tex]\cfrac{(x-2+x+2)(x-2-x-2)}{(x+2)(x-2)}=[/tex]          Factorize the numerator

[tex]\cfrac{2x(-4)}{(x+2)(x-2)}=[/tex]                                   Simplify the numerator

[tex]-\cfrac{8x}{x^2-4}[/tex]                                                Answer

Used identity:

The equation to find the entrance cost is 5(c + 3) = 85.

Given that 5 five friends made plan for trip, which will cost them a total of $85,

This includes a $3 fee added to each entrance ticket to cover the cost of a pair of grip socks.

We need to establish an equation to find c, the cost of an entrance ticket,

So,

The total cost of one entrance ticket = c + 3,

So, for 5 people = 5(c+3)

Now, this is equal to 85.

So the equation is = 5(c + 3) = 85.

Hence the equation to find the entrance cost is 5(c + 3) = 85.

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The X-Coordinate of 3 only represents a specific moment in time during the orbit and not necessarily the speed of the ISS at all times.

Since the graph is showing how fast the International Space Station (ISS) is traveling in orbit around the Earth, the x-coordinate of 3 likely represents a specific point in time or duration of the orbit.

To find out what the x-coordinate of 3 represents, we need to know the units of time on the x-axis. If the units are in minutes, then a value of 3 would represent 3 minutes. Similarly, if the units are in hours, then a value of 3 would represent 3 hours.

Assuming the units on the x-axis are in minutes, the point with an x-coordinate of 3 represents the speed of the ISS 3 minutes into its orbit. This means that at that point in time, the ISS was traveling at a speed of approximately 27,500 kilometers per hour.

It's important to note that the speed of the ISS can vary throughout its orbit, depending on factors such as altitude, atmospheric conditions, and changes in the orbit itself. Therefore, the x-coordinate of 3 only represents a specific moment in time during the orbit and not necessarily the speed of the ISS at all times.

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Note the full question is

Representing Proportional Relationships

This graph shows how fast the International Space Station travels in orbit around Earth.

What is the meaning of the point with an x-coordinate of 3?

In order to plot points H I and J on the coordinate planeIdentify the x-coordinate and y-coordinate of each point (H, I, and J). For example, let's say the coordinates are as follows:

Point H: (xH, yH)

Point I: (xI, yI)

Point J: (xJ, yJ)

Locate the origin (0, 0) on the coordinate plane. This is the starting point for plotting any point.

Move along the x-axis according to the x-coordinate of each point. If the x-coordinate is positive, move to the right; if it's negative, move to the left. Mark a point on the x-axis at the appropriate location.

Move along the y-axis according to the y-coordinate of each point. If the y-coordinate is positive, move upward; if it's negative, move downward. Mark a point on the y-axis at the appropriate location.

The point where the x and y axes intersect is the location of the point. Mark that point on the coordinate plane.

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How to Plot points H I and J on the coordinate plane

The missing length indicated has the value given as follows:

x = 25.

The geometric mean theorem states that the length of the altitude drawn from the right angle of a triangle to its hypotenuse is equal to the geometric mean of the lengths of the segments formed on the hypotenuse.

The bases in this problem are given as follows:

x and 144.

The length of the altitude segment is given as follows:

60.

60 is the geometric mean of x and 144, hence the value of x is obtained as follows:

144x = 60²

x = 3600/144

x = 25.

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The correct statement is:
As x → ∞, f(x) → ∞ and as x → -∞, f(x) → -∞.

I understand that you are asking about the behavior of the function g(z) as x approaches positive and negative infinity. Let's analyze the given function:
g(z) = 4x³ - 3r + 2r³ - 1
First, I believe "r" should be "x" to keep the variables consistent. So, the corrected function is:
g(x) = 4x³ - 3x + 2x³ - 1
Now, let's find the end behavior of this function as x approaches positive and negative infinity.
As x → ∞:
The highest degree term, x³, will dominate the function's behavior. The coefficients for x³ are 4 and 2, so the function will behave like 6x³. Since this term is positive and has an odd exponent, as x → ∞, f(x) → ∞.
As x → -∞:
Again, the x³ term will dominate the function's behavior. Since the exponent is odd, as x → -∞, f(x) → -∞.
So, the correct statement is:
As x → ∞, f(x) → ∞ and as x → -∞, f(x) → -∞.

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Answer:

≈226.1947

Step-by-step explanation:

have a great day and thx for your inquiry :)

Answer:

To determine how the golfer ended in terms of par, tally up all of her scores and deduct them from the total par for the seven rounds.

The total par for seven rounds multiply by eighteen is 126 (assuming the course is a regular par 72).

The golfer's score in relation to par for each round is:

1 round at 3 over par = 75 (72 + 3)

2 rounds with 1 over par = 73 (72 + 1) x 2 = 146

3 rounds with 3 under par = 69 (72 - 3) x 3 = 207

1 round at 1 under par = 71 (72 - 1)

you add all the pars working above, which is

75 + 146 + 207 + 71 = 499

Her score of 499 is 27 below par (126 - 499 = -373), meaning she ended significantly below par.

Using the ASA Triangle Congruence Theorem, we can prove that triangles GHJ and PQR are congruent because they have Angle G and angle P are congruent, both being right angles.

The two triangles are congruent by rigid motions.

Using the ASA Triangle Congruence Theorem, we can prove that triangles GHJ and PQR are congruent because they have:

Angle G and angle P are congruent, both being right angles.

Side GH and side PQ are congruent, both having a length of 5 units.

Angle H and angle Q are congruent, both having a measure of 63.43 degrees (rounded to two decimal places).

Since the two triangles have two congruent angles and a congruent side between them, they are congruent by the ASA Triangle Congruence Theorem.

b. We can also prove that triangles GHJ and PQR are congruent using rigid motions.

Specifically, we can use a translation followed by a reflection to map triangle GHJ onto triangle PQR.

Translation: We can translate triangle GHJ 2 units to the left and 3 units down to get triangle G'H'J', where G'(-2, 1), H'(-2, -2), and J'(2, -2).

Reflection: We can reflect triangle G'H'J' across the x-axis to get triangle PQR, where P(-2, -4), Q(-2, -1), and R(2, -1).

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The utilization factor can be calculated as the ratio of the arrival rate to the service rate. In this case, the arrival rate (lambda) is 15 per hour, and the service rate (mu) is 25 per hour. Therefore, it is important for the clinic to monitor the utilization factor and adjust staffing levels as needed to ensure efficient and effective patient communication.

So the utilization factor is:
Utilization factor = lambda / mu
Utilization factor = 15 / 25
Utilization factor = 0.6
This means that on average, the operator is utilized 60% of the time. In other words, there is a 40% idle time during which no phone calls are being processed. This is a relatively low utilization factor, which suggests that there is some capacity to handle more phone calls if necessary. However, it is important to note that if the arrival rate were to increase, the utilization factor would increase as well, which could lead to longer wait times for patients and potential bottlenecks in the phone system.

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Answer:

1/2 ㏑(sec (2x + 9)) + c  ........ absolute sec (2x +9)

Step-by-step explanation:

∫tan (2x + 9) dx

let u = 2x + 9

du/dx = 2

dx = du/2 = (1/2) du

now we have ∫tan u (1/2  du)

= 1/2 ∫tan u du

= 1/2 ㏑ (sec u) ...... absolute

=1/2 ㏑(sec (2x + 9)) + c ........ absolute

The perimeter of the triangle is 73 unit.

We know that tangents drawn from external point are equal in length.

So, HE = EI = 16 unit

IF = GF = 9.5 unit

DG = DH = 11 unit

So, the length of sides are

DE = 16 + 11 = 27

EF = 9.5 + 16 = 25.5

FD = 11 + 9.5 = 20.5

Now, the perimeter of Triangle DEF is

= DE + EF + FD

= 27 + 25.5 + 20.5

= 73 unit

Therefore, the perimeter of the triangle is 73 unit.

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To find the volume of liquid needed to fill a sphere of radius R to a height of R/5, we can consider the spherical cap formed by the liquid.

The volume of a spherical cap can be calculated using the formula:

V = (1/3)πh²(3R - h)

Where V is the volume, h is the height of the cap, and R is the radius of the sphere.

In this case, the height of the cap is R/5 and the radius of the sphere is R. Substituting these values into the formula, we get:

V = (1/3)π(R/5)²(3R - R/5)

= (1/3)π(R²/25)(15R - R/5)

= (1/3)π(R²/25)(74R/5)

= (74/75)πR³

So, the volume of liquid needed to fill the sphere to a height of R/5 is (74/75)πR³.

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To obtain a mixture of coffee that can be sold for $60 a pound, the merchant must mix the $20 coffee with the $80 coffee in such a way that the resulting mixture is worth $60 per pound.

To determine how many pounds of the $20 coffee should be used, we can set up a system of equations using the weights of each type of coffee and their corresponding values per pound.

Let x be the number of pounds of the $20 coffee to be used. Then, the total weight of the mixture is 80 + x pounds. The value of the $20 coffee is $20 per pound, and the value of the $80 coffee is $80 per pound. The value of the mixture must be $60 per pound. Therefore, we can write the following equation:

$20x + 80(80) = 60(80 + x)$

Simplifying and solving for x, we get:

$x = 200$

Therefore, the merchant should use 200 pounds of the $20 coffee and mix it with 80 pounds of the $80 coffee to obtain a mixture that can be sold for $60 a pound.

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Answer:

If x < -12, then (-x) > 12, and we have:

x - (-12) = x + 12

So, the simplified expression without using the absolute value symbol is:

x + 12

Step-by-step explanation:

Answer:

Since x is less than -12, then x-(-12) = x+12.

Here is a table of values for x and x-(-12):

x | x-(-12)

-13 | -1

-14 | -2

-15 | -3

...

Step-by-step explanation:

The percentile rank of a score of 141, to the nearest percentile, is 53.

The percentile rank describes the percentage of scores in the frequency distribution that are equal to or lower than the achieved score.

To calculate the percentile rank of a score of 141, we can proceed as follows:

Firstly, the scores are arranged in ascending order: 88, 95, 104, 110, 111, 112, 112, 120, 123, 134, 141, 145, 152, 156, 166, 166, 168, 177, 177, 190.

The number of scores below 141 = 10

The number of scores equal to 141 = 1

The following percentile rank formula can be used to determine the percentile rank of a score of 141:

Percentile Rank = [(M + (0.5 * R)) / Y] x 100, where:

M = Number of Ranks below x

R = Number of Ranks equals x

Y = Total Number of Ranks

Percentile Rank = [(10 + (0.5 * 1)) / 20] x 100

= 52.5

Percentile Rank = 53

Thus, the percentile rank of a score of 141 in a test with a maximum possible score of 200 is 53.

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The volume of the right circular cone with a height of 15.4 m and a base circumference of 18 m is 15.68 cubic meters.

To find the volume of a right circular cone, we can use the formula:

Volume = (1/3) × π × r² × h,

where π is the mathematical constant pi, r is the radius of the base, and h is the height of the cone.

Given that the base has a circumference of 18 m, we can calculate the radius using the formula:

Circumference = 2 × π × r

18 = 2 × π × r

r = 18 / (2 × π) = 2.864 m.

Now we can substitute the values into the volume formula:

Volume = (1/3) × π × (2.864 m)² × 15.4 m,

Volume = 15.68 m³

Therefore, the volume of the right circular cone with a height of 15.4 m and a base circumference of 18 m is 15.68 cubic meters.

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Answer:

The correct equation is D. 13x - 2y = 40

13(2) - 2(-7) = 26 + 14 = 40

13(4) - 2(6) = 52 - 12 = 40

The equation will be y = kx²z², where k is the proportionality constant.

Given that y varies jointly as the square of x, the square of z,

And when x = 3 and z = 4, then y = 72,

So, we can write,

y ∝ x²z²

or,

y = kx²z² [k = proportionality constant]

Now, given x = 3 and z = 4, then y = 72,

Therefore,

72 = 3²·4²·k

72 = 9·16 k

k = 72 / 144

k = 0.5

Therefore, the value of proportionality constant k is 0.5.

Hence the equation will be y = kx²z², where k is the proportionality constant.

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Let's first start by rewriting the denominator to have a base of 3.

9 = 3^2

9^(n - 1) = 3^2(n - 1)

Now that we have two terms with the same base, we can subtract the exponents.

3^n / 3^2(n - 1) = 3^[n - 2(n - 1)]

3^[n - 2n + 2]

3^[-n + 2]

3^-(n - 2)

Answer: 3^(-n + 2) OR 3^-(n - 2)

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The surface area of the triangular prism in square meters is:

549.15 m²

The total surface area of the given prism would be the sum of the areas of the individual faces that make up the prism.

The formula for area of a triangle is:

Area = ¹/₂ * base * height

The formula for area of a rectangle is:

Area = Length * width

Thus:

Total surface area of prism is:

TSA = (15 * 13) + (15.81 * 15) + 2(¹/₂ * 9 * 13)

TSA = 549.15 m²

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The probability of choosing exactly two $50 tickets and no other money winners is:

34,320 / 21,455 ≈ 0.

we can use the hypergeometric    probability   distribution to solve this problem, since we are drawing without replacement from a finite population of tickets with different values.

the total number of tickets in the box is:

13 + 11 + 11 + 20 = 55

the number of ways to choose exactly two $50 tickets from the 13 available is:

${13 \choose 2} = 78$

the number of ways to choose one dummy ticket from the 20 available is:

${20 \choose 1} = 20$

the number of ways to choose one ticket from the remaining non-$50 tickets is:

11 + 11 = 22

, the total number of ways to choose exactly two $50 tickets and no other money winners is:

78 x 20 x 22 = 34,320

the number of ways to choose any three tickets from the 55 available is:

${55 \choose 3} = 21,455$ 60 or about 60%

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The equation of the line represented by the points on the graph is  (5x+16)/6.

line is a straight one-dimensional figure that does not have a thickness, and it extends endlessly in both directions.

To calculate the equation of the line represented by the points on the graph, we use the formula below

Formula:

From the points in the graph,

Given:

Substitute these values into equation 1

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Answer:  

[tex]\displaystyle \frac{8}{3} \;\text{feet per picture frame}[/tex]

Step-by-step explanation:

      We will divide the total amount of trim that Francois has, 8 feet, by the number of picture frames, 3. This will give us a fraction that represents how many feet of trim can be used for each picture frame.

       [tex]\displaystyle \frac{\text{8\;\;feet}}{\text{3 picture frames}} =\frac{8}{3} \;\text{feet per picture frame}[/tex]

The relative speed between the jogger and the walker during the 500 second interval is 2.24 m/s.

To solve this problem, we need to first find the distances traveled by each person.

The jogger travels 10 blocks x 100 meters/block = 1000 meters south.

The walker travels 5 blocks x 100 meters/block = 500 meters west.

Using the Pythagorean theorem, we can find the distance between them: √((1000m)² + (500m)²) = 1118.03 meters.

To find their relative speed, we divide this distance by the time interval: 1118.03 meters / 500 seconds = 2.24 m/s.

Therefore, their speed relative to each other during this interval is 2.24 m/s.

The jogger traveled 1000 meters south, while the walker traveled 500 meters west. Using the Pythagorean theorem, the distance between them is 1118.03 meters. Dividing this distance by the time interval of 500 seconds, we get their relative speed, which is 2.24 m/s.

: The relative speed between the jogger and the walker during the 500 second interval is 2.24 m/s.

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Prove of expression tan² θ - sin² θ = tan² θ × sin² θ is shown below.

Given that;

Expression is,

⇒ tan² θ - sin² θ = tan² θ × sin² θ

Now, We can simplify as;

⇒ tan² θ - sin² θ

⇒ sin² θ / cos² θ - sin²θ

⇒ sin² θ (1 / cos² θ - 1)

⇒ sin² θ (sec² θ - 1)

⇒ sin² θ × tan² θ

Thus, Prove of expression tan² θ - sin² θ = tan² θ × sin² θ is shown.

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Remember - SOH CAH TOA

cos(A) = adjacent / hypotenuse

adjacent = 5

sin(A) = opposite / hypotenuse

opposite = 12

hypotenuse = 13

tan(A) = opposite / adjacent

tan(A) = 12 / 5

Answer: tan(A) = 12/5

Hope this helps!