Part A
Study the two functions shown, A(t) and 12∙J(t). Based on the graph and the data, what kinds of functions are they? Choose among linear, quadratic, and exponential. Describe the features of each function that gave you clues.







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Part B
The equation A = Pert describes a bank loan that compounds continuously. The variables in the equation are described in the table:

Variable Definition
A This is the principal and interest on the loan. Principal is the amount of money borrowed. Interest is a graduated fee paid to the bank for the privilege of borrowing its money.
P This is the principal, or the amount of money borrowed. Don’t confuse P in the compounding interest equation with P in the profit equation. One is principal, the other is profit.
e This is Euler’s number, e ≈ 2.7, used in exponential functions that are continuously compounding.
r This is the interest rate expressed as a percentage.
t This is the time allotted, in years, to repay the loan. It’s also called the life of the loan.
For the sake of this activity, assume that you will collect profit from sales for a number of months and then use a portion of that profit to pay off the entire loan in one lump-sum payment once the loan terminates. Based on this assumption, what does the intersection of the 12∙J(t) curve and the A(t) curve represent? Explain using your own words.






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Part C
Take some time to gradually increase P in increments of $100,000 while keeping r and J(t) constant. What happens to the relationship between the two curves? What does this mean with respect to the bank loan? Why is this a dangerous situation with respect to the financial health of your business? Why would banks put safeguards in place to prevent this from occurring?







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Answer:

x+4y=7

Step-by-step explanation:

The value of the unknown side to the nearest hundredth is 6.78

Cosine rule states that; if a, b,c are the sides of a triangle and A,B,C are the opposite angles of the sides, then

c² = a² + b² -2ab CosC

Cosine rule is used when all the sides or two of the sides are given. The angle opposite to side c is angle C.

c² = 4² + 10² - 2(4)(10)cos 29

c² = 16 +100 - 80cos29

c² = 116 -69.97

c² = 46.03

c = √ 46.03

c = 6.78 ( nearest hundredth).

Therefore the value of c is 6.78

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John's z-score is 0.75.

John's score of 79 on the math test corresponds to a z-score of 0.75 and a percentile of 77.82%.

To get John's z-score, we will use the formula: z = (x - μ) / σ

Data

x = John's score (79)

μ = mean score (73)

σ = standard deviation (8)

z = (79 - 73) / 8

z = 6 / 8

z = 0.75.

To find John's percentile, we can use a standard normal distribution table or calculator.

The percentile represents the percentage of scores that fall below a given value. Using  standard normal distribution table, we find that a z-score of 0.75 equals to percentile of 77.82%.

Full question:

The scores on a math test have a mean of 73 points and a standard deviation of 8 points. John scores a 79 on the test. Convert this score to a z-score and a percentile..

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The cumulative frequency to be used in solving the 28th percentile is 16.

Cumulative frequency is described as  the total of a frequency and all frequencies in a frequency distribution until a certain defined class interval.

we add up the frequencies until we reach or exceed 28% of the total data set in order to find the cumulative frequency for the 28th percentile.

Total frequency = 4 + 9 + 16 + 10 + 9 + 2 = 50

28th percentile = (28/100) * Total frequency

28th percentile = (28/100) * 50 = 14

We start from the beginning of the distribution and add up the frequencies until we reach or exceed 14 so as to find the cumulative frequency.

In conclusion, and with reference to the given data set we notice that the cumulative frequency exceeds 14 at the 91-120 minute interval = 16.

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Answer:

5

She must therefore earn 5 points on the last test to achieve a mean score of 4.

Step-by-step explanation:

The sum of the data divided by the total number of data represents the mean, which is the average of a set of data.

Answer:

$1250

Step-by-step explanation:

A = P + Prt

1337.5 = P + P(0.04)(21/12)

1337.5 = 1.07P

P = 1250

Answer: $1250

The total tax due for a married couple who have a combined income of $110,000 is $16,079

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

Income = $110,000

Also, we have the table of values

From the tax table, we have the category of tax to be paid by the couple to be

Tax = 8907 + 22% * excess of 77400

So, we have

Tax = 8907 + 22% * (110,000 - 77400)

Evaluate

Tax = 16079

Hence, the tax amount is $16,079

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Answer:

[tex]8 {x}^{2} - 6x - 5 = x[/tex]

[tex]8 {x}^{2} - 7x - 5 = 0[/tex]

x = (7 + √((-7)^2 - 4(8)(-5)))/(2×8)

= (7 + √(49 + 160))/16

= (7 + √209)/16

= -.4661, 1.3411 (to 4 decimal places)

Answer:

Mean: 27

Median: 22

Mode: 22

Standard deviation: 17

No outlier (61 might be one just check with people you know and see if they have an outlier or not)

Step-by-step explanation: Give a definition of mean, median, mode…and just add your answers.

The area of the rectangle is 1296 square feet.

Let's denote the width of the rectangle as "w" and its length as "4w" since the length is four times the width.

The formula for the perimeter of a rectangle is given by:

P = 2(l + w), where P is the perimeter, l is the length, and w is the width.

Given that the perimeter of the rectangle is 180 ft, we can write the equation as:

180 = 2(4w + w)

Simplifying the equation:

180 = 2(5w)

180 = 10w

w = 18

So, the width of the rectangle is 18 ft.

Now, we can find the length of the rectangle:

Length = 4w = 4(18) = 72 ft

The area of a rectangle is given by the formula:

[tex]A = l \times w,[/tex]

where A is the area,

l is the length,

and w is the width.

Substituting the values we found:

Area [tex]= 72 ft \times 18 ft = 1296 ft^2[/tex]

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Answer:

Look Below

Step-by-step explanation:

The statement "definition of finiteness uses the notion of a natural number" means that when defining what it means for a set or collection to be finite, the concept of a natural number is employed.

In mathematics, a set is said to be finite if it can be put into a one-to-one correspondence with a specific subset of the natural numbers. This subset typically starts with the number 1 and includes a finite sequence of consecutive natural numbers, depending on the size of the set.

For example, consider a set A with three elements: A = {a, b, c}. To show that A is finite, we can establish a one-to-one correspondence between the elements of A and the set {1, 2, 3} as follows: a ↔ 1, b ↔ 2, c ↔ 3. This mapping shows that each element of A corresponds to a unique natural number, and vice versa. Since there is a bijection (a one-to-one correspondence) between the elements of A and a subset of the natural numbers, we conclude that A is finite.

Thus, in the definition of finiteness, the notion of a natural number is used to provide a precise and rigorous characterization of what it means for a set to be finite. It allows us to establish a clear distinction between finite sets and infinite sets, which do not have a one-to-one correspondence with any subset of the natural numbers.

Answer:

42

Step-by-step explanation:

Rectangle: A = 6 x 5 = 30

Triangle: A = 1/2(6 x 4) = 12

Area of figure: 30 + 12 = 42

Answer:

area of rectangle+area of triangle= area of figure

6×5 + 1/2×4×6=30+24= 42

or

given figure is a trapezium

so we know are of trapezium= 1/2×sum of parallel sides×height

= 1/2×( 5+9)×6

=42

Answer:

42 = 23 + j

Step-by-step explanation:

The equation representing the given sentence would be:

42 = 23 + j

Here, the variable "j" represents Janelle's age.

There are 19 nickels, 15 quarters, and 18 dimes in the collection.

Let's assume that there are n number of nickels in the collection.There are four fewer quarters than nickels, so the number of quarters will be n - 4.

There are 3 more dimes than quarters, so the number of dimes will be (n - 4) + 3 = n - 1.

The value of each nickel is $0.05, the value of each dime is $0.10, and the value of each quarter is $0.25.Using this information, we can form the following equation to represent the total value of the collection:

0.05n + 0.10(n - 1) + 0.25(n - 4) = 6.5Simplifying this equation, we get:0.05n + 0.10n - 0.10 + 0.25n - 1 = 6.50.40n - 1.10 = 6.50

Adding 1.10 to both sides, we get:0.40n = 7.60Dividing both sides by 0.40, we get:n = 19

Therefore, there are 19 nickels in the collection.There are four fewer quarters than nickels, so the number of quarters will be n - 4 = 19 - 4 = 15.

There are 3 more dimes than quarters, so the number of dimes will be (n - 4) + 3 = 15 + 3 = 18.

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The solution of this equation is z = 3.

We are given that;

1-3+4z = 1

Now,

To solve for z, we need to isolate it on one side of the equation. We can do this by following these steps:

Subtract −31​

from both sides to eliminate it from the left side.

Multiply both sides by z to eliminate the fraction on the left side.

Divide both sides by 4 to isolate z on the left side.

Let’s see how this works:

−31​+z4​=1

−31​+z4​=1

z4​=1+31​

z4​=34​

4=4/3 ​z

z=3*4​/4​

z=4/4×3​

z=3

Therefore, by the equation answer will be z = 3.

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y = 3x - 1: consistent, independent

2y - 6x = 4: consistent, independent

2y = -x + 6: consistent, independent

4y + 2x = 12: consistent, independent

y = 2x + 3: consistent, independent

x + y = -9: consistent, dependent

y = 3x - 1 and 2y = -x + 6: inconsistent

Let's match each system on the left with the appropriate words that describe the system on the right:

y = 3x - 1: consistent, independent

This system represents a linear equation with a unique solution. The equations are not multiples of each other, and they intersect at a single point.

2y - 6x = 4: consistent, independent

Similar to the first system, this represents a linear equation with a unique solution. The equations are not multiples of each other, and they intersect at a single point.

2y = -x + 6: consistent, independent

This system also represents a linear equation with a unique solution. The equations are not multiples of each other, and they intersect at a single point.

4y + 2x = 12: consistent, independent

Like the previous systems, this represents a linear equation with a unique solution. The equations are not multiples of each other, and they intersect at a single point.

y = 2x + 3: consistent, independent

This system represents a linear equation with a unique solution. The equations are not multiples of each other, and they intersect at a single point.

x + y = -9: consistent, dependent

This system represents a linear equation with infinitely many solutions. The equations are multiples of each other, meaning they represent the same line. Therefore, any point on the line satisfies both equations.

y = 3x - 1 and 2y = -x + 6: inconsistent

This system represents inconsistent equations that do not have a common solution. The lines represented by the equations are parallel and do not intersect.

To summarize the matches:

y = 3x - 1: consistent, independent

2y - 6x = 4: consistent, independent

2y = -x + 6: consistent, independent

4y + 2x = 12: consistent, independent

y = 2x + 3: consistent, independent

x + y = -9: consistent, dependent

y = 3x - 1 and 2y = -x + 6: inconsistent

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Answer: Unfactorable

Step-by-step explanation:

5v^2-30v+70

5(v^2-6v+14)

You can't factor v^2-6v+14 with rational numbers

Subtraction and Division

Inverse operations help find the solution to equations.

Defining Inverse Operations

Firstly, let's define an operation. An operation in math is a function that can manipulate a value. Inverse operations are operations that are opposite operations that undo each other. For example, addition and subtraction are inverse operations because subtraction undoes addition. Multiplication and division are also inverse operations.

Solving the Equation

The equation 16 = 3x + 1 involves both addition and multiplication. So, to solve this, we can use the inverse operations of subtraction and division. First, subtract 1 from both sides.

Then, divide both sides by 3.

This shows that by using subtraction and division, we can undo the addition and multiplication used in the equation. This allows us to find the value of x.

The expression that shows the volume of the prism is x³ - x² - 6.

The prism above is a rectangular prism. The volume of the prism can be found as follows:

The volume of the prism can be found as follows:

volume of the rectangular prism = lwh

where

Therefore,

volume of the rectangular prism = (x - 3)(x)(x + 2)

volume of the rectangular prism = (x² - 3x)(x + 2)

volume of the rectangular prism = x³ + 2x² - 3x² - 6

volume of the rectangular prism = x³ - x² - 6

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The length of segment OP, which represents the distance from the origin to point P on the unit circle, is always equal to 1.

To determine the length of segment OP on the unit circle, we need to use trigonometry. Let's break down the problem step by step:

Definition: The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system.

Initial Ray: The initial ray is a line segment that starts from the origin (0, 0) and extends to a point on the unit circle. It forms an angle with the positive x-axis.

Rotation: We are rotating the initial ray counterclockwise by θ degrees. This means we are essentially finding a new point on the unit circle based on the angle θ.

Trigonometric Functions: The trigonometric functions sine (sin) and cosine (cos) are particularly useful for calculating the coordinates of points on the unit circle.

sin(θ) gives the y-coordinate of a point on the unit circle.

cos(θ) gives the x-coordinate of a point on the unit circle.

Coordinates of Point P: Since we are rotating the initial ray counterclockwise by θ degrees, the coordinates of point P on the unit circle can be obtained as follows:

x-coordinate of P: cos(θ)

y-coordinate of P: sin(θ)

Distance from the Origin (Length of Segment OP):

Using the coordinates of point P, we can calculate the distance between the origin (0, 0) and point P using the distance formula.

The distance formula states that for two points (x1, y1) and (x2, y2), the distance between them is given by:

d = √((x2 - x1)² + (y2 - y1)²)

In this case, point P has coordinates (cos(θ), sin(θ)), and the origin is (0, 0). Thus, the distance (length of segment OP) is:

d = √((cos(θ) - 0)² + (sin(θ) - 0)²)

= √(cos²(θ) + sin²(θ))

= √(1) [Using the trigonometric identity: sin²(θ) + cos²(θ) = 1]

= 1

Therefore, the length of segment OP, which represents the distance from the origin to point P on the unit circle, is always equal to 1.

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Answer:

At the time the ball was thrown, it was 8 feet above the ground.

h'(x) = -32x + 64 = 0, so x = 2

The ball reaches its maximum height after 2 seconds.

The general form of the solutions of the recurrence relation is

[tex]a_n = (A + B n + C n^2 + D n^3) 1^n + (E + F n + G n^2) (-2)^n + (H + I n) 3^n + (J + K n) (-4)^n,[/tex]

We are given that;

Series=1, 1, 1, 1, -2, -2, -2, 3, 3, -4

Now,

In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought.

If the characteristic equation has repeated roots, then the general solution is modified by multiplying each term by a polynomial in n whose degree is equal to the multiplicity of the root minus one. For example, if r is a root of multiplicity m, then the corresponding term in the general solution is

[tex]a_n = (A + B n + C n^2 + … + Z n^(m-1)) r^n,[/tex]

where A, B, C, …, Z are arbitrary constants.

In this question, the characteristic equation has roots 1, 1, 1, 1, -2, -2, -2, 3, 3, -4. The root 1 has multiplicity 4, the root -2 has multiplicity 3, and the roots 3 and -4 have multiplicity 2 each.

[tex]a_n = (A + B n + C n^2 + D n^3) 1^n + (E + F n + G n^2) (-2)^n + (H + I n) 3^n + (J + K n) (-4)^n,[/tex]

where A, B, C, D, E, F, G, H, I, J and K are arbitrary constants.

Therefore, by the equation answer will be [tex]a_n = (A + B n + C n^2 + D n^3) 1^n + (E + F n + G n^2) (-2)^n + (H + I n) 3^n + (J + K n) (-4)^n,[/tex]

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The algebraic expression for the sentence in this problem is given as follows:

n - (x + 8).

The sentence in this problem is given as follows:

"A number n minus the sum of the number and eight".

The sum of an unknown number x and eight is given as follows:

x + 8.

The subtraction of the number n by the expression is given as follows:

n - (x + 8).

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Answer:

y = -1/2x

Step-by-step explanation:

Currently, y = 2x - 5 is in slope-intercept form, whose general equation is y = mx + b, where

The slopes of perpendicular lines are negative reciprocals of each other.  We can show this with the following formula;

m2 = -1 / m1, where

Finding the slope of the other line:

Since 2 is the slope of y = 2x - 5, we can plug in 2 for m1 in perpendicular slope formula to find m2, the slope of the other line:

m2 = -1 / 2

m2 = -1/2

Thus, the slope of the other line is m = -1/2.

Finding the y-intercept of the other line:

We can find the y-intercept of the other line (i.e., b in the slope-intercept equation) by plugging in (-2, 1) for x and y and -1/2 for m in the slope-intercept form:

1 = -1/2(-2) + b

1 = 1 + b

0 = b

Thus, y = -1/2x is the line that passes through (-2, 1) and is perpendicular to y = 2x - 5.

Since all three points satisfy the equation 2x + 2y = 24 when their coordinates are substituted, we can conclude that the points (4,8), (2,10), and (5,7) indeed lie on the line represented by the equation 2x + 2y = 24.

To determine if the points (4,8), (2,10), and (5,7) lie on the line 2x + 2y = 24, we can substitute the x and y values of each point into the equation and check if the equation holds true.

For the point (4,8):

2(4) + 2(8) = 8 + 16 = 24, which satisfies the equation.

For the point (2,10):

2(2) + 2(10) = 4 + 20 = 24, which also satisfies the equation.

For the point (5,7):

2(5) + 2(7) = 10 + 14 = 24, which satisfies the equation as well.

Since all three points satisfy the equation 2x + 2y = 24 when their coordinates are substituted, we can conclude that the points (4,8), (2,10), and (5,7) indeed lie on the line represented by the equation 2x + 2y = 24.

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Using the division operation , Natalie can do 64 loads of laundry with her bottle of detergent.

From the line plot , the detergent with 4 bottles has 8 cups in it.

Fraction of cup used per laundry = 1/8

Total cups of detergent in bottle = 8

Number of laundry Natalie can do :

(Total cups of detergent in bottle / Fraction of cup used per laundry )

Number of loads of laundry that can be done :

= 8 / (1/8)

= 8 × 8/1

= 64

Therefore, Natalie can do 64 loads of laundry with her bottle of detergent.

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The completely factorized forms are:

1.1) 2x + 4y - 6z (no further factorization)

[tex]1.2) 2pq(5p^5q - 2p^2 + 1)[/tex]

1.3) (m + n)(m + n - 5p)

1.4) 7c(4 - a) + d(-4 + a)

We have,

Let's factorize each expression completely:

1.1)

2x + 4y - 6z

There are no common factors among the terms, so we can't factorize it further.

1.2)

[tex]10p^6q^2 - 4p^3q^2 + 2pq[/tex]

The common factor among the terms is 2pq, so we can factor it out:

[tex]2pq (5p^5q - 2p^2 + 1)[/tex]

1.3)

(m + n)² - 5p(m + n)

Using the distributive property, we can expand the squared term:

(m + n)(m + n) - 5p(m + n)

Now we can factor out the common factor (m + n):

(m + n)(m + n - 5p)

1.4)

4(7c - d) + a(d - 7c)

Using the distributive property, we can expand the terms:

28c - 4d + ad - 7ac

Rearranging the terms:

(28c - 7ac) + (-4d + ad)

Factoring out common factors:

7c(4 - a) + d(-4 + a)

Thus,

The completely factorized forms are:

1.1) 2x + 4y - 6z (no further factorization)

[tex]1.2) 2pq(5p^5q - 2p^2 + 1)[/tex]

1.3) (m + n)(m + n - 5p)

1.4) 7c(4 - a) + d(-4 + a)

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A triangular prism is a three-dimensional geometric shape that consists of two triangular bases and three rectangular faces connecting them. It is a polyhedron with six faces, nine edges, and six vertices.

The two triangular bases of a triangular prism are congruent and parallel to each other. The rectangular faces are perpendicular to the triangular bases, and their lateral edges connect the corresponding vertices of the triangular bases.

The triangular bases are identical and parallel to each other. Each base has three vertices, three edges, and one face.There are three rectangular lateral faces connecting the corresponding vertices of the triangular bases. Each lateral face has two edges and one face.

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The new coordinates of the quadrilateral ABCD after the dilation is:

A' = (-6, 6)

B' = (-8, -4)

C' = (4, -8)

D' = (0, 0)

We have,

To find the coordinates of the new quadrilateral after a dilation with a scale factor of 2 centered at the origin, we multiply the x and y coordinates of each point by the scale factor.

Given the original coordinates of the quadrilateral ABCD:

A = (-3, 3)

B = (-4, -2)

C = (2, -4)

D = (0, 0)

To dilate the coordinates with a scale factor of 2, we multiply each coordinate by 2:

A' = 2 x (-3, 3) = (-6, 6)

B' = 2 x (-4, -2) = (-8, -4)

C' = 2 x (2, -4) = (4, -8)

D' = 2 x (0, 0) = (0, 0)

Thus,

The new coordinates of the quadrilateral ABCD after the dilation is:

A' = (-6, 6)

B' = (-8, -4)

C' = (4, -8)

D' = (0, 0)

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