Question 2(Multiple Choice Worth 2 points)
(Linear Functions MC)
Which of the following tables represents a linear function?
x1 1 011
y-2-1012
x0 123 4
y-320-23
x-2-10 12
y 2 0
-102
x-2-10 1 2
y 3 1
-1-3-5

The original matrix [ 1 5 −3 h −2 −7] and [ 2 4 3 6 h 7]  is the augmented matrix of a consistent linear system, regardless of the value of h.

The augmented matrix of a consistent linear system, we need to perform row operations and transform it into an echelon form or a reduced row echelon form. In this form, we can easily read the system of linear equations and determine its consistency.

Let's apply some row operations to the given matrix. We can start by dividing the first row by 2 to obtain a leading 1 in the first column:

[ 1 2 3/2 3 h/2 7/2]

Next, we can subtract twice the first row from the second row to get a zero in the first column of the second row:

[ 1 2 3/2 3 h/2 7/2]

[ 0 1 -3/2 0 h/2 -5/2]

Now, we can subtract 3/2 times the second row from the third row to get a zero in the first column of the third row:

[ 1 2 3/2 3 h/2 7/2]

[ 0 1 -3/2 0 h/2 -5/2]

[ 0 0 9/2 3 -3h/4 13/4]

At this point, the matrix is in echelon form, and we can read the system of linear equations:

x + 2y + (3/2)z = 3

y - (3/2)z + (h/2)w = -5/2

(9/2)z + 3w - (3h/4)v = 13/4

Therefore, the original matrix [ 2 4 3 6 h 7] is the augmented matrix of a consistent linear system, regardless of the value of h.

Exercise 22 follows a similar procedure. The matrix is [ 1 5 −3 h −2 −7], and we can apply row operations to transform it into echelon form:

[ 1 5 -3 h -2 -7]

[ 0 1 -1/5 h/5 4/5 -3/5]

[ 0 0 -6/5 (h²)/5 -38/25 22/25]

The resulting system of linear equations is:

x + 5y - 3z + hw - 2v = -7

y - (1/5)z + (h/5)w + (4/5)v = -3/5

(-6/5)z + (h²)/5w - (38/25)v = 22/25

We see that the last row has no zeros, except for the rightmost column, so the system is consistent for any value of h

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The surface area of the cylinder has a radius of 9.4cm and a height of 18.6 cm is 1098.56 square centimeters.

The surface area of this cylinder is defined as the product of the radius of the base and the height of the cylinder.

The surface area of this cylinder = (2π x r)h

We have given a radius of the base is 9.4 cm and the height of the cylinder is 18.6 cm.

The surface area of this cylinder = (2π x r)h

= (2π x 9.4)18.6

= (59.06)18.6

= 1098.56

Thus, The surface area of the cylinder has a radius of 9.4cm and a height of 18.6 cm is 1098.56 square centimeters.

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Measure of angle G is 62.5°.

For a regular polygon of any number of sides, the sum of its exterior angle is 360° (full angle).

In triangle EFG,

Measure of angle E = 55°

Measure of angle F should be less than 180 = x/2°

By the triangle sum theorem, the Sum of measures of all interior angles of triangle = 180°

m(∠E) + m(∠F) + m(∠G) = 180°

55° + x/2° + x/2= 180°

x= 180° - 55°

x = 125

Measure of angle G = 125/2 = 62.5

Therefore, measure of angle G is 62.5°.

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In the figure, angles ∠5 and ∠1 are complementary angle.

The sum of two or more angle is 90 degree then the angle is called the complementary angle.

Given that;

Angles are shown in figure.

Now, By the figure we have;

⇒ ∠ 1 and ∠ 4 are vertically opposite angles, hence both are equal.

Here, ∠4 and ∠5 are complementary angles.

⇒ ∠4 + ∠5 = 90°

⇒ ∠1 + ∠5 = 90°

Hence,  Angles ∠5 and ∠1 are complementary angle.

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The big-o for the expected work performed by a call to fib(n) (where n < maxmemo) will be constant.

If the function is implemented with memoization, i.e., storing the results of previously calculated Fibonacci numbers to avoid redundant calculations, the expected work performed by a call to fib(n) would be O(1) for n < maxmemo, regardless of the number of previous calls made to fib.

This is because if the function has already calculated fib(n) earlier, it can simply look up the result in the memoized dictionary and return the stored value without performing any additional calculations. Otherwise, it would calculate fib(n) once and store the result in the memoized dictionary for future lookups.

Therefore, the expected work performed by a call to fib(n) is constant, O(1), as long as the previously calculated Fibonacci numbers are stored in memory using memoization. The number of previous calls made to fib has no impact on the expected work performed by a call to fib(n) as long as n is less than maxmemo.

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Diagonals of a parallelogram bisect each other.

[tex]x+29=7x+5 \implies x=4 \\ \\ 3y=8y-35 \implies y=7[/tex]

The amount of interest that Laura will be paid in the first 5 years is $600.

The simple interest formula is expressed as;

I = P × r × t

Where I is interest, P is principal, r is interest rate and t is time.

Given that;

Plug the given values into the above formula and solve for interest I.

I = P × r × t

I = $3,000 × 0.04 × 5

I = $3,000 × 0.2

I = $600

Therefore, the amount of interest payable is $600.

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Divide the 8-digit student ID by the leftmost 3 digits to find q and r using the Quotient Remainder Theorem. If your 8-digit student ID is 12345678, then d = 123, n = 45,678, q = 371, and r = 45.

Suppose your 8-digit understudy ID is 12345678. Then, at that point, we can take the furthest left three digits to get d = 123. Also, we can take the excess 5 digits to get n = 45,678.

To find q and r, we can utilize the condition n = dq + r and address for q and r.

To begin with, we really want to see as the biggest various of d that is not exactly or equivalent to n. We can do this by separating n by d and taking the floor of the outcome:

q = ⌊n/d⌋ = ⌊45,678/123⌋ = 371

Then, we can track down the rest of taking away the result of d and q from n:

r = n - dq = 45,678 - 123 × 371 = 45,678 - 45,633 = 45

Along these lines, in the event that your 8-digit understudy ID is 12345678, d = 123, n = 45,678, q = 371, and r = 45.

The Remainder Leftover portion Hypothesis expresses that for any whole number n and positive whole number d, there exist whole numbers q and r with the end goal that n=dq+r, where q is the remainder and r is the rest of. We can track down q and r by separating n by d and taking the floor of the outcome for q, and deducting the result of d and q from n for r.

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The solution of the modulus equation is -

x = 2

x = - 2

Given is a function as -

|2x| = - 4

We can write -

For {x < 0} → {|x| = - x}. We can write the function as -

- 2x = - 4                                

x = 2

For {x > 0} → {|x| = x}. We can write the function as -

2x = - 4

x = - 2

Therefore, the solution of the modulus equation is -

x = 2

x = - 2

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The post-neonatal mortality rate for the year is 1.1×10⁻⁷.

The post-neonatal mortality rate in a year is applicable when the death day lies between 28 and 364. First, divide the number of post-neonatal deaths by the number of live births. Then multiply the obtained value by 1000 to get the post-neonatal mortality rate.

It is known that 1 billion units = 10⁹ units. So, 20 billion live births will be 20×10⁹ live births. Also, the number of post-neonatal deaths is 2200.

So, use the formula, post-neonatal mortality rate = (the number of post-neonatal deaths)/(the number of live births).

post-neonatal mortality rate = 2200/(20×10⁹)

= 110×10⁻⁹

= 1.1×10⁻⁷

Therefore, the obtained answer is 1.1×10⁻⁷.

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The flour is needed for a big batch of cookies that uses 24 cups of oatmeal would be 20 cups.

The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.

In order to find out how many groups of 6 cups of oatmeal there are.

So, For every 6 cups of oatmeal = 24 / 6 = 4

Now,

For 5 cups of flour = 4 x 5 = 20

The flour is needed for a big batch of cookies that uses 24 cups of oatmeal would be 20 cups.

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Answer:

Step-by-step explanation:

35 + 2x

Answer:

xy+2x

x=7

y=5

input the value of x and y in the equation.

7×5+(2×7)

=35+14

=49

The number of ordered triples (a, b, c) where a, b, and c are positive integers and a is a factor of b, a is a factor of c, and abc=N, is 64.

To find the ordered triplets, Firstly Factorize N into its prime factors.For each prime factor. we can choose its exponent as the value of a, b, and c. The total number of ordered triples (a, b, c) is (number of exponents + 1)^3.

let N = 24. The prime factorization of 24 is 2^3 * 3^1.

The number of possible values of a is 4 (from 0 to 3), so the number of possible values of b and c is also 4.

The number of ordered triples is 4^3 = 64.

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The solution is, the required probability is p= 1/10.

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.

here, we have,

from the given information we get,

We have 5 possible parts colored red, green, blue, purple and yellow

The cube for this case have 6 possible numbers 1,2,3,4,5,6

We have the following even numbers: 2,4,6

Then the probability of obtain an even number is:

p1= 3/6= 1/2

Then the probability of obtain a blue color on the spinner is:

p2= 1/5

Then we want this probability:

Obtain an even number AND sppining a blue on the spinner

Then since the events are independent we can do this:

p = 1/2 * 1/5= 1/10

Hence, The solution is, the required probability is p= 1/10.

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Invested in each vehicle is $40,000 if the annual investment income is $944.25.

Let

CD's = x

Bonds = x + 60000

Stocks = y

x + x + 60000 + y = 140000 ----> 2x + y = 80000 .......(1)

Using I = Prt; t = 1; I = Pr

2x + 3.7(x + 60000) + 11.1y = 7800(100× )  × Since I used an exact proportion, the opposite side needs to be multiplied by 100.

5.7x + 11.1y = 558000 .......(2)

Either elimination or substitution;

x = 20000 y = 40000

So CD's = 20000, Bonds = 80000, Stocks = 40000

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Comparing the z-scores, we can see that Sean's SAT score is slightly higher relative to the mean and standard deviation of the SAT population than his ACT score is relative to the mean and standard deviation of the ACT population.

To find the z-score for Sean's SAT score, we use the formula:

z = (x - μ) / σ

where x is Sean's SAT score, μ is the population mean (1500), and σ is the population standard deviation (250). Substituting the values, we get:

z = (1700 - 1500) / 250 = 0.8

Therefore, Sean's z-score on the SAT is 0.8.

To find the z-score for Sean's ACT score, we use the formula:

z = (x - μ) / σ

where x is Sean's ACT score, μ is the population mean (20.4), and σ is the population standard deviation (4.8). Substituting the values, we get:

z = (24 - 20.4) / 4.8 = 0.75

Therefore, Sean's z-score on the ACT is 0.75.

Comparing the z-scores, we can see that Sean's SAT score is slightly higher relative to the mean and standard deviation of the SAT population than his ACT score is relative to the mean and standard deviation of the ACT population.

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The Triangle ABC has been dilated through point p and scale factor 3 to produce this finished picture.

Having three sides, three angles, and three points, a triangle is indeed a closed, two-dimensional object. A hexagon also includes a triangular. Triangles can be found in a variety of everyday objects, such as sandwiches, traffic signals, clothes hooks, and billiards racks.

Assume point P as origin then.

P has coordinate (0,0).

And point A has coordinate (-4,0).

point B has coordinate (0,-3).

point C has coordinate (-4,-3).

P(0,0), A(-4,0) B(0,-3) C(-4,-3).

Since the triangle goes an dilation with center  at P and scale factor 3.

Then k=scale factor =3.

Now the new coordinate of Triangle is given by A',B' and C', whose coordinate in (x, y) plane is calculated as.

[tex](\mathrm{x}, \mathrm{y}) \rightarrow(\mathrm{kx}, \mathrm{ky})[/tex]

Now calculating the Coordinates on dilated triangle

A'=3(-4,0)=(-12,0)

B'=3(0,-3)=(0,-9)

C'=3(-4,-3)=(-12,-9).

We will calculate the coordinates of dilated triangle assuming the point P as origin

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Answer:

b = -9

Step-by-step explanation:

-3 (3b + 6) = 36 - 3b

-9b - 18 = 36 - 3b

-6b - 18 = 36

-6b = 54

b = -9

Let's check

-3(3(-9) + 6) = 36 - 3(-9)

-3( -27 + 6) = 36 + 27

-3( -21) = 63

63 = 63

So, b = -9 is the correct answer.

Step-by-step explanation:

-3(3b+6)=36-3b

-9b-18=36-3b

-9b+3b=36+18

-6b=54

divide both sides by -6

b= -9

To determine how many cakes Lionel makes in one hour, we can use the following formula:

cakes per hour = total cakes / total hours worked

On Monday, Lionel made 7 cakes in some number of hours, so we don't know how many cakes he made per hour that day. However, on Tuesday, he made an additional 6 cakes (13 total cakes - 7 cakes made on Monday), and he worked for 2 hours longer than he did on Monday. Therefore:

cakes per hour on Tuesday = 6 cakes / 2 hours = 3 cakes per hour

So Lionel made 3 cakes per hour on Tuesday. We don't have enough information to determine how many cakes he made per hour on Monday.

The shape of this histogram is skewed to the right since the highest frequency is found in the first class, and the frequencies decrease as the numbers increase.

The class width, also known as the class interval, is the difference between the upper and lower boundaries of a class. In this frequency distribution, the class width is 9 since the lower boundary of the first class is 54.5 and the upper boundary of the last class is 90.5. We can calculate the frequency of each class by counting how many numbers fall within the class boundaries. For example, the first class has 72 observations since there are 72 numbers that fall between 54.5 and 63.5. To calculate the relative frequency of each class, we divide the frequency of each class by the total frequency. For example, the relative frequency of the first class is 0.312 since 72/231 = 0.312. Once the frequencies and relative frequencies have been calculated, a histogram can be made to represent the data. The shape of this histogram is skewed to the right since the highest frequency is found in the first class, and the frequencies decrease as the numbers increase.

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The value of c is 1/15

Joint probability is a statistical measure that calculates the likelihood of two events occurring together and at the same point in time. Joint probability is the probability of event Y occurring at the same time that event X occurs.

To be valid probability functions, the sum of changes across the entire range of x and y variables must equal one

b) In this case, the joint probability could be represented as follows:

                     x = -2                  x = 0                    x = 2

y = -2               0                         2c                        4c

y = 3                5c                       3c                          c

As a consequence, the total likelihood in this case is:

=> c(2 + 4 + 5 + 3 + 1) = 15c

=> 15c = 1

c = 1/15

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Answer:

False.

Step-by-step explanation:

It is not possible to create a regular tessellation with a regular heptagon. A regular tessellation refers to a pattern of repeating shapes that completely covers a flat surface without any gaps or overlaps. The only regular polygon shapes that can form regular tessellations are the equilateral triangle, square, and hexagon. Regular heptagons do not have the symmetry necessary to form a repeating pattern that covers a flat surface without gaps or overlaps.

Pythagorean theorem and its converse to prove that triangle ABC is a right triangle.

The converse of Pythagorean theorem states that if the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle.

Let's begin by visualizing the given triangle ABC and the line CN that is perpendicular to side AB. Since CN is perpendicular to AB, we can say that triangle ACN and triangle BCN are right triangles (as the angle at C is 90 degrees).

Now, let's focus on the point H that is the geometric mean of points D and E. This means that DH multiplied by HE is equal to CH squared (as per the definition of geometric mean).

We can represent this relationship using the Pythagorean theorem as well. Since triangle CDH and triangle CEH are right triangles (as angle C is 90 degrees), we can say that:

CD² + DH² = CH² (using Pythagorean theorem for triangle CDH)

CE² + EH² = CH² (using Pythagorean theorem for triangle CEH)

From the given relationship of geometric mean, we can substitute CH² in the above equations and get:

CD² + DH² = HE*DH

CE² + EH² = HE*EH

Adding these two equations, we get:

CD² + CE² + DH² + EH² = HE*(DH + EH)

AB² = 4HE² (using the fact that DH + EH = DE = 2HE)

This equation gives us an important insight about the triangle ABC. It tells us that the sum of the squares of the sides (AB²) is equal to four times the square of the length HE (4HE²).

Now, we can use the converse of Pythagorean theorem to prove that triangle ABC is a right triangle.

In our case, we have AB² = 4HE², which satisfies the converse of Pythagorean theorem. Therefore, we can conclude that triangle ABC is a right triangle.

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The best definition of the unit rate is how much of something per 1 unit of something else. Hence option D is the correct answer

Unit rate is basically a ratio and it compares something with 1 unit of some other quantity. So they are used for comparing two different things and if we talk about unit rate then the denominator is always 1 or the other value is always 1

So if we talk about an example of unit rate and if we specifically talk about speed, then we say kilometer per hour

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area of square 1 is 28 square units and its side length is[tex]$2\sqrt{7}$[/tex] units.

Let the side length of square 1 be x units. Then the areas of squares 1, 2, and 3 are [tex]$x^2$[/tex], 64, and 36 square units, respectively.

Since square 2 has an area of $64$ square units, its side length is [tex]$\sqrt{64}=8$[/tex] units. Similarly, square 3 has a side length of[tex]$\sqrt{36}=6$[/tex]units.

We can now set up an equation based on the fact that the sum of the areas of squares 1 and 3 is equal to the area of square 2:

[tex]$$x^2+36=64$$[/tex]

Solving for x, we get:

[tex]$$x^2=28$$[/tex]

Therefore, the area of square 1 is [tex]$x^2=28$[/tex] square units, and its side length is [tex]$\sqrt{28}=2\sqrt{7}$[/tex] units.

In summary, the area of square 1 is 28 square units and its side length is[tex]$2\sqrt{7}$[/tex] units.

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Looking at the derivative, we see that it is defined for all values of t, so there are no vertical tangents.

The critical spots where the slope of the curve is zero or undefinable must be located in order to identify the points of horizontal tangency.

Our first step is to determine y's derivative with regard to x:

dy/dx = 3t^2 - 0

In the event that the derivative is set to zero, we obtain:

3t^2 = 0

t = 0

We gain one crucial insight from this at (22, 0).

The places where the derivative is undefinable must be located in order to determine the vertical tangent. There are no perpendicular tangents when looking at the derivative because it is defined for all values of t.

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The solution is, $1.33, is the cost per quart.

In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.

here, we have,

given that,

A drum of oil contains 55 gal and costs $164.80.

now, we know , 1 gal = 4 quart

so, 55 gal =55 * 4

                 = 220 quart

now, the cost per quart= 220/164.80

                                      =$1.33

Hence, The solution is, $1.33, is the cost per quart.

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The linear function relationship between the change in the median home value and the year, indicates that we get;

a) The home values increased at a higher rate in Connecticut

b) The median home value in West Virginia in 2010 is $80,721

c) The home values in both states are equal in 1915

A linear function relationship can be expressed in the form; y = m·x + c. Where, m is the slope of the graph of the function and c is the y-intercept.

a) The rate of increase of the home value, where the values of the house change linearly can be found as follows;

Rate of change of home value for West Virginia =  [tex]\frac{72800-33200}{200-1950} = 792[/tex]

The home values in West Virginia increases by about $792 per year

Rate of change of home values for Connecticut = [tex]\frac{166900-71900}{2000-1950} =1900[/tex]

The home values in Connecticut increases by $1900 each year

Therefore;

b) The increase in the home value in West Virginia per year is $792, therefore, the home value in 2010 can be found as follows;

The number of years between 2010 and 2000 = 2010 - 2000 = 10

Home value in 2010 = Home value in 2000 + 792 × 10

Home value in 2010 = $72800 + $792 × 10 = $80720

c) The equation for the median home value for West Virginia can be presented as follows;

y - 33200 = 792×(x - 1950)

y = 792×(x - 1950) + 33200

The equation for the median home value for Connecticut can be presented as follows;

y - 71900 = 1900×(x - 1950)

y = 1900×(x - 1950) + 71900

When the home values in West Virginia is equal to the home value in Connecticut, we get;

792 × (x - 1950) + 33200 = 1900 × (x - 1950) + 71900

1900·x - 792·x = 1900 × 1950 - 792 × 1950 + 33200 - 71900 = 2121900

1108·x = 2121900

x = 2121900/1108 ≈ 1915.07

The year the median value will be the same in both states is about 1915

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