model with a proportion
The decrease in price for notebooks was $1.60, a 40% decrease. What was al original price and what is the new price?

Answer:

There are a total of 15 customers, so the probability of selecting any one customer at random is 1/15.

If the first customer selected is a Pike resident, there are 6 Pike residents remaining out of a total of 14 customers remaining. So the probability of the second customer being a Pike resident, given that the first customer was a Pike resident, is 6/14.

Therefore, the probability of both customers selected being Pike residents is:

(6/15) * (6/14) = 0.1714 or approximately 0.17 (rounded to two decimal places).

So the probability that both customers selected are Pike residents is approximately 0.17.

Answer:

$6236.56

Step-by-step explanation:

Compound Formula

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

I plugged in the numbers of P, r, and t.

n means the number of times the interest is applied per period. It is compounded monthly, and the time is given 3 years.

This means 12 months x 3 years = 36 months.

[tex]A = $5700(1+.03/36)^3^6^(^3^)[/tex]

[tex]A=6236.559672[/tex]

[tex]A=6236.56[/tex]

The binomial theorem holds for any integer \( n \geq 1 \) and any real numbers \( x \) and \( y \).

The binomial theorem states that for any positive integer \( n \) and any real numbers \( x \) and \( y \), \[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k\]where \(\binom{n}{k} = \frac{n!}{k!(n-k)!} \) is the binomial coefficient.

To prove the identity \[ \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k = x^n + nx^{n-1}y + \frac{n(n-1)}{2}x^{n-2}y^2 + \cdots + y^n \] we can use differentiation, integration, and multiplication by \( x \) or \( y \).

First, let's differentiate both sides of the equation with respect to \( x \): \[\frac{d}{dx} \left( \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k \right) = \frac{d}{dx} \left( x^n + nx^{n-1}y + \frac{n(n-1)}{2}x^{n-2}y^2 + \cdots + y^n \right)\]Using the power rule for differentiation, we get \[\sum_{k=0}^{n} \binom{n}{k}(n-k)x^{n-k-1}y^k = nx^{n-1} + n(n-1)x^{n-2}y + \frac{n(n-1)(n-2)}{2}x^{n-3}y^2 + \cdots\]Next, we can multiply both sides of the equation by \( x \): \[\sum_{k=0}^{n} \binom{n}{k}(n-k)x^{n-k}y^k = nx^{n} + n(n-1)x^{n-1}y + \frac{n(n-1)(n-2)}{2}x^{n-2}y^2 + \cdots\]Finally, we can integrate both sides of the equation with respect to \( x \): \[\int \left( \sum_{k=0}^{n} \binom{n}{k}(n-k)x^{n-k}y^k \right) dx = \int \left( nx^{n} + n(n-1)x^{n-1}y + \frac{n(n-1)(n-2)}{2}x^{n-2}y^2 + \cdots \right) dx\]Using the power rule for integration, we get \[\sum_{k=0}^{n} \binom{n}{k}\frac{(n-k)}{n-k+1}x^{n-k+1}y^k = \frac{n}{n+1}x^{n+1} + \frac{n(n-1)}{n+1}x^{n}y + \frac{n(n-1)(n-2)}{2(n+1)}x^{n-1}y^2 + \cdots\]Simplifying the coefficients and combining like terms, we get \[\sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k = x^n + nx^{n-1}y + \frac{n(n-1)}{2}x^{n-2}y^2 + \cdots + y^n\]which is the identity we were trying to prove. Therefore, the binomial theorem holds for any integer \( n \geq 1 \) and any real numbers \( x \) and \( y \).

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Answer:

81.55

Step-by-step explanation:

because 4 twentydollar bills = 80 dollars 1 five-dollar bill = 5.00

80 + 5 = 85 dollars 2 quarts = 50 cents and 1 dime equals 10 cent 50 + 10 = 60 together you have 85.60 when you subtract 4.05 minus 85.60 you get 81.50 cent

The quotient is [tex]x^{3}[/tex]+2[tex]x^{2}[/tex]+7x+20 and the remainder is 62.

To find the quotient and remainder using synthetic division, we can follow these steps:

1. Write down the coefficients of the dividend, which are 1, -1, 1, -1, and 2.

2. Write down the value of x from the divisor, which is 3.

3. Bring down the first coefficient, 1, to the bottom row.

4. Multiply the value of x, 3, by the first coefficient in the bottom row, 1, and write the result, 3, in the second column of the top row.

5. Add the second coefficient in the dividend, -1, to the value in the second column of the top row, 3, and write the result, 2, in the second column of the bottom row.

6. Repeat steps 4 and 5 for the remaining columns.

7. The bottom row will contain the coefficients of the quotient, and the last value in the bottom row will be the remainder.
The synthetic division will look like this:
3|1-11-12|362160|1272062

Therefore, the quotient is  [tex]x^{3}[/tex]+2[tex]x^{2}[/tex]+7x+20 and the remainder is 62. The final answer is ([tex]x^{3}[/tex]+2[tex]x^{2}[/tex]+7x+20)+62 / (x-3).

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A poset on [54] = {1, 2, ... ,54} with comparisons a < b if and only if a divides b is a partially ordered set where elements are related if one divides the other. The height of this poset is 6 (1, 2, 4, 8, 16, 32, 54), and the width is 10 (all the elements in the set).

A poset on [54] is a partially ordered set that is defined with the comparison a < b if and only if a divides b. In this poset, the elements are ordered based on the divisibility relation, meaning that an element a is considered to be less than another element b if and only if a divides b.

The height of this poset is the maximum number of elements in a chain, which is 6. This can be seen by considering the chain {1, 2, 4, 8, 16, 32}.

The width of this poset is the maximum number of elements in an antichain, which is 10. This can be seen by considering the antichain {3, 5, 6, 7, 10, 14, 15, 21, 22, 35}.

Therefore, the height and width of this poset are:
Height = 6
Width = 10

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The sum of the circumferences of the two circles is equal to [tex]$2\pi \sqrt 2.$[/tex]

Circumference is the distance around a two-dimensional shape, such as a circle or ellipse. It can be calculated by multiplying the circumference of the shape by its diameter. The formula for calculating the circumference of a circle is 2πr, where π is the constant 3.14 and r is the radius of the circle. The circumference of an ellipse is more complicated and requires knowledge of the length of its major and minor axes.

The two circles are externally tangent to each other, which means that the distance between them is equal to the sum of their radii. Since the circles are tangent to the sides of the square, the length of one side of the square is equal to the sum of their radii. Since the perimeter of the square is given to be [tex]$2+\sqrt 2,[/tex] we can calculate the length of each side of the square to be [tex]$\sqrt 2.$[/tex] Hence, the sum of the radii of the two circles is equal to [tex]$\sqrt 2.$[/tex]
Therefore, the sum of the circumferences of the two circles is equal to [tex]$2\pi \sqrt 2.$[/tex]

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Step-by-step explanation:

21

Table that represent input & output value  would be:

Input Output

44            45

45            46

46            47

To determine which table represents inputs and outputs that follow the same rule as 44+1, 44+2, and 44+3, we can simply evaluate each of the expressions and look for a pattern.

44 + 1 = 45

44 + 2 = 46

44 + 3 = 47

Therefore, the rule appears to be adding 1 to each subsequent number. We can check this by evaluating additional terms:

44 + 4 = 48

44 + 5 = 49

Based on this pattern, the correct table would be:

Input Output

44            45

45            46

46            47

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The coordinates of A' and B' after a dilation with a scale factor of 2 with a center point of dilation at the origin is: OA (-8, 2) and B (-2, 6)

To find the coordinates of A' and B' after a dilation with a scale factor of 2 with a center point of dilation at the origin, we can use the following formula:

A' = k * (A - O) + O

B' = k * (B - O) + O

where k is the scale factor, O is the center point of dilation, and A and B are the original points.

In this case, k = 2 and O = (0, 0). So we have:

A' = 2 * (-4, 1 - (0, 0)) + (0, 0) = (-8, 2)

B' = 2 * (-1, 3 - (0, 0)) + (0, 0) = (-2, 6)

Therefore, the coordinates of A' and B' after the dilation are (-8, 2) and (-2, 6), respectively.

So, the answer is option (A) OA (-8, 2) and B (-2, 6).

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First, we need to convert the maximum weight that the mail carrier is allowed to carry from pounds to ounces:

30 pounds = 30 x 16 ounces = 480v ounces

Then, we can divide the maximum weight by the weight of one letter:

480 ounces / 12 ounces per letter = 40 letters

Therefore, the mail carrier is able to carry 40 letters.

So, the correct answer is option C: 40 letters.

Answer:

C 40

Step-by-step explanation:

By substituting the extreme points of the feasible set into the objective function and comparing the values, (x,y)=(1,1) is an optimal solution for the given LP problem.

The given LP problem is to maximize f(x,y)=3x+19y subject to the constraint set of the feasible set. The extreme points of the feasible set are (0,0), (1,0), (0,1), and (1,1).

To prove that (x,y)=(1,1) is an optimal solution, we need to show that f(x,y) is maximized at this point. We can do this by plugging in the extreme points into the objective function and comparing the values.

At (0,0), f(x,y) = 3(0) + 19(0) = 0

At (1,0), f(x,y) = 3(1) + 19(0) = 3

At (0,1), f(x,y) = 3(0) + 19(1) = 19

At (1,1), f(x,y) = 3(1) + 19(1) = 22

From these calculations, we can see that f(x,y) is maximized at (1,1), with a value of 22. Therefore, (x,y)=(1,1) is an optimal solution for this LP problem.

In conclusion, by plugging in the extreme points of the feasible set into the objective function and comparing the values, we have proved that (x,y)=(1,1) is an optimal solution for the given LP problem.

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The area of the following circle is A ≈ 153.86 square units.

A circle is a two-dimensional geometric shape that consists of all the points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius, and the distance across the circle passing through the center is called the diameter.

The circumference of a circle is the distance around the edge of the circle, and it is calculated using the formula C = 2πr, where r is the radius and π (pi) is a mathematical constant approximately equal to 3.14159. The area of a circle is the region enclosed by the circle, and it is calculated using the formula A = πr².

The diameter of the circle is 14, so the radius is half of that, which is 7.

The area of the circle is given by the formula A = πr², where r is the radius. Substituting in the values we get:

A = π(7)²

A = 49π

Therefore, the area of the circle is 49π square units. If you want to use an approximation, you can use 3.14 as an estimate for π and get:

A ≈ 153.86 square units.

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The complete question is -

The equation that describes the plant's height is y = 3 + 0.5x .

The equation that describes the plant's height is a linear equation. A linear equation is an equation that has a single variable that is raised to the power of one.

The general form of a linear equation is:

y = mx + c

Where:

m = slope

c = intercept

Plant's height = starting height + (rate of growth x number of weeks)

y = 3 + (0.5 x x)

y = 3 + 0.5x

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The location of the corner of the rabbit pen that reflects (2.5, 2.5) across the y-axis is (-2.5, 2.5).

The coordinate plane, also called the Cartesian plane, is a two-dimensional grid used to locate points in space. It consists of two perpendicular number lines, one horizontal and one vertical, that intersect at their zero points, called the origin.

Points on the coordinate plane are identified by their coordinates, which are written as ordered pairs of numbers (x, y), where x is the horizontal coordinate and y is the vertical coordinate.

To solve this problem, we need to reflect the point (2.5, 2.5) across the y-axis. When we reflect a point across the y-axis, we keep the y-coordinate the same but change the sign of the x-coordinate.

So, the reflected point will have the same y-coordinate of 2.5, but the x-coordinate will be -2.5. Therefore, the location of the corner of the rabbit pen that reflects (2.5, 2.5) across the y-axis is (-2.5, 2.5).

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Answer: 18

Step-by-step explanation

​

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 27x, the least common multiple of 27,x.

x × 6=27 × 4

Multiply 27 and 4 to get 108.

x × 6=108

Divide both sides by 6.

x= 108/6

​

Divide 108 by 6 to get 18.

x=18

Note that the pair of numbers that solves the system of equations:

y=11x+40; and

y= -7x+9

are,  x = -31/18  ; and y = 379/18.

To solve the system of equations, we need to find the values of x and y that satisfy both equations simultaneously. One way to do this is to set the expressions for y equal to each other, since both expressions represent the same value of y.

So we have:

11x + 40 = -7x + 9

Simplifying and solving for x, we get:

18x = -31

x = -31/18

Now that we have found the value of x, we can substitute it into either equation to find the corresponding value of y. Let's use the first equation:

y = 11(-31/18) + 40

y = -341/18 + 720/18

y = 379/18

Therefore, the pair of numbers that solves the system of equations is (-31/18, 379/18).

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The vectors are linearly independent.

The vectors u = (2,1,0), v = (-4,3,1), and w=(0,-2,-5) are linearly dependent if there exists scalars c1, c2, and c3 such that c1u + c2v + c3w = 0. To determine if this is the case, we can form a matrix with the vectors as columns and find its determinant. If the determinant is 0, then the vectors are linearly dependent. If the determinant is not 0, then the vectors are linearly independent.

The matrix formed by the vectors is:
```
| 2 -4 0 |
| 1  3 -2|
| 0  1 -5|
```

The determinant of this matrix is:
```
2(3(-5) - (-2)(1)) - (-4)(1(-5) - (0)(-2)) + 0(1(1) - (0)(3))
= 2(-15 + 2) + 4(5) + 0
= 2(-13) + 20 + 0
= -26 + 20
= -6
```

Since the determinant is not 0, the vectors are linearly independent.

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The required polynomial in standard form for the volume of the cylinder is [tex]$48\pi x^3 + 64\pi x^2 + 24\pi x + 4\pi$[/tex].

The formula for the volume of a cylinder is [tex]$V = \pi r^2 h$[/tex], where r is the radius and h is the height.

In this case, the radius is given as 4x + 1, and the height is given as 3x + 4. So we can substitute these values into the formula to get:

[tex]$$V = \pi(4x + 1)^2(3x + 4)$$[/tex]

Simplifying the expression inside the parentheses first, we have:

[tex]$$(4x + 1)^2 = (4x + 1)(4x + 1) = 16x^2 + 8x + 1$$[/tex]

Substituting this expression into the formula for V, we get:

[tex]$$V = \pi(16x^2 + 8x + 1)(3x + 4)$$[/tex]

Expanding the expression using the distributive property, we get:

[tex]$$V = \pi(48x^3 + 64x^2 + 24x + 4)$$[/tex]

Simplifying further, we get:

[tex]$$V = 48\pi x^3 + 64\pi x^2 + 24\pi x + 4\pi$$[/tex]

Therefore, the polynomial in standard form for the volume of the cylinder is [tex]$48\pi x^3 + 64\pi x^2 + 24\pi x + 4\pi$[/tex].

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Ticket sales fell $960 below the goal.

What is system of equations?

A system of linear equations can be solved graphically, by substitution, by elimination, and by the use of matrices.

Since we know that the goal is to raise $3,300 each night and that the price of an adult ticket is $7.50 and the price of a student ticket is $4.50, we can write:

7.5d + 4.5s = 3300

We also know that the auditorium holds 600 people, so the total number of tickets sold must be:

d + s = 600

total number of tickets sold was 480. We can use this information to set up another system of equations:

s = 3d (since there were three times as many student tickets sold as adult tickets)

d + s = 480 (since the total number of tickets sold was 480)

Now we can solve the first system of equations to find the values of d and s that satisfy the constraints:

7.5d + 4.5s = 3300

d + s = 600

Multiplying the second equation by 4.5 and subtracting it from the first equation, we get:

3d = 1650

So, d = 550. Substituting this value back into the equation d + s = 600, we get:

550 + s = 600

s = 50

Therefore, 550 adult tickets and 50 student tickets must have been sold to raise exactly $3,300.

To answer the second part of the question, we can use the second system of equations to find the values of d and s for that performance:

s = 3d

d + s = 480

Substituting the first equation into the second equation, we get:

d + 3d = 480

So, 4d = 480 and d = 120. Substituting this value back into the first equation, we get:

s = 3d = 360

Therefore, 120 adult tickets and 360 student tickets were sold at that performance.

To calculate how much below the goal of $3,300 ticket sales fell, we can plug in the values for d and s from this performance into the equation:

7.5d + 4.5s = revenue

7.5(120) + 4.5(360) = $2,340

So, ticket sales fell $960 ($3,300 - $2,340) below the goal.

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The spinner  probability that the next spin will land either red or green or purple is found 0.70 as a decimal to the nearest hundredth.

Spinner Results

Color Frequency

Red 20

Blue 9

Green 19

Yellow 14

Purple 14

Total outcomes: 20 + 9 + 19 + 14 + 14

Total outcomes: 76

spinner probability = favorable outcome / total outcome

favorable outcome (red or green or purple) = 20 + 19 + 14 = 53

spinner probability = 53 / 76 = 0.69

spinner probability = 0.70

Thus, the probability that the next spin will land either red or green or purple is found 0.70 as a decimal to the nearest hundredth.

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The given equation y = (1/2)x -2 represents a linear function because it has an independent and a dependent variable, each with an exponent of 1.

A linear function is defined as an equation in which the highest exponent of the variable is always one.

To determine the equation "y equals one-half times x minus 2" represents a linear or nonlinear function.

The algebraic form of this phrase "y equals one-half times x minus 2" is :

y = (1/2)x -2

Since it has an independent "x" and a dependent variable "y" with an exponent of one, the above equation y = (1/2)x -2 defines a linear function.

Therefore, the correct answer would be an option (C).

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The value of angle e is 55⁰.

The value of angle f is 55⁰.

The value of angle e is calculated by applying the following principles of angles on a straight line.

The vertical angle between e⁰ and 100⁰ = 25⁰ ( vertically opposite angles are equal)

The value of angle e is calculated as;

e⁰ + 25⁰ + 100⁰ = 180⁰ ( sum of angles on a straight line )

e⁰ = 180⁰ - 125⁰

e⁰ = 55⁰

The missing base angle of the triangle on the same line as e is calculated as;

? + 110⁰ = 180⁰ (sum of angles on a straight line )

? = 180 - 110

? = 70⁰

The value of angle f is calculated as;

f⁰ + e⁰ + ? = 180⁰ ( sum of angles in a triangle )

f⁰ + 55⁰ + 70⁰ = 180

f⁰ = 180 - 125⁰

f⁰ = 55⁰

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Answer: 6 cm

solution let the length of AB= x cm the length of BC = (2x-2) cm, and the length of AC = (x+10) cm The perimeter of ABC=32 cm

x+2x-2+x+0=32

4x+8=32

4x=24

x=6

The perimeter of the given triangle MNP is 65.

A regular figure with n-sides has n equal sides in it, and they are the only parts of it(that means, nothing more than those equal lengthened n sides).

Suppose that length of each side of that figure be of u units, then we have the perimeter as:

P=u+u+u+u+u+u......=n*u

units.

We are given that;

Side MN=5x-34, QR=25, QS=22, RS=x+4

Now,

5x + x - 34 + 4 = 22

6x - 30 = 22

6x - 30 + 30 = 22 + 30

6x = 52

6x/6 = 52/6

x = 8.67

P= 5*8-34+25+22+8+4

=40-34+47+12

=65

Therefore, the perimeter of the triangle will be 65.

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The answer is g(-2) = 15.

To find g(-2) using synthetic division, we need to follow these steps:

Write down the coefficients of the polynomial in descending order. The coefficients are 11, 29, 19, 22, and 39.

Write down the value of the divisor in the leftmost column. In this case, the divisor is -2.

Bring down the first coefficient (11) to the bottom row.

Multiply the bottom row value (11) by the divisor (-2) and write the result (-22) in the next column.

Add the coefficient in that column (29) to the result (-22) and write the sum (7) in the bottom row.

Repeat steps 4 and 5 until you reach the last column. The final result in the bottom row is the remainder.

The synthetic division should look like this:

-2 | 11  29  19  22  39
   | -22 -14 -10 -24
   -------------------
     11   7   9  12  15

Therefore, g(-2) = 15.

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The area of the dark region is 164.15 square feet because it is equal to the sum of the areas of the circles and rectangles.

In geometry, a circle is a closed object made up of all the points in a plane that are equally spaced from the circle's centre. The radius and diameter of a circle are measured from the centre to any spot on the circle, respectively. The circumference, which is equivalent to pi times the diameter, is the distance around the circle.

given

[tex]area of circle = π * r * r[/tex]

= 22/7 * 2 * 2 = 88/7 ft2

area of rectangle = length * breadth

= 9.4 * 18.8

= 176.72 ft2

area of shaded region

176.72 - 12.57

= 164.15 ft 2

The area of the dark region is 164.15 square feet because it is equal to the sum of the areas of the circles and rectangles.

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Answer:

Step-by-step eEIKxplanation:

The dimension of a vector space is equal to the number of linearly independent vectors in its basis.

In the given basis B = {(0,2,1),(0,2,1),(1,0,0)}, we can see that the first two vectors (0,2,1) and (0,2,1) are identical and therefore not linearly independent. Therefore, the dimension of the vector space is equal to the number of linearly independent vectors in the basis, which is 2.

So, the dimension of the vector space whose basis are B = {(0,2,1),(0,2,1),(1,0,0)} is equal to 2.

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