Justin, Cam, and Ben are playing a board game where exactly one player will win. Ben estimates that Justin has a
20
%
20%20, percent chance of winning each game and that Cam has a
50
%
50%50, percent chance of winning each game.

The price of one teddy bear is $7 and the price of one doll is $14.

Let's use matrices to solve this system of equations:

First, we need to define the variables:

x = price of one teddy bear

y = price of one doll

Then we can write the system of equations:

14x + 3y = 158

8x + 12y = 296

system of matix:

| 14   3 |   | x |   | 158 |

|  8  12 | * | y | = | 296 |

To solve for x and y, we can use matrix multiplication and inversion:

| x |   | 12  -3 |   | 158 |   |  99 |

| y | = | -8  14 | * | 296 | = | -14 |

So, x = $7 and y = $14. Therefore, the price of one teddy bear is $7 and the price of one doll is $14.

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The sum of the series under the interval (3, 6) will be negative 4.

Given that:

Series, ∑ (2k - 10)

A series is a sum of sequence terms. That is, it is a list of numbers with adding operations between them.

The sum of the series under the interval (3, 6) is calculated as,

∑₃⁶ (2k - 10) = (2 x 3 - 10) + (2 x 4 - 10) + (2 x 5 - 10) + (2 x 6 - 10)

∑₃⁶ (2k - 10) = - 4 - 2 + 0 + 2

∑₃⁶ (2k - 10) = -4

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There are 3 ordered pairs (x, y) that satisfy the equation y=(x-5)^2-1.

To find the ordered pairs (x, y) for 3 < x < 7 that satisfy the equation y=(x-5)^2-1, follow these steps:

Step 1: Set the range of x values: 3 < x < 7

Step 2: Plug in each whole number value of x within the given range (4, 5, and 6) into the equation and calculate the corresponding y values.

For x = 4:
y = (4 - 5)^2 - 1
y = (-1)^2 - 1
y = 0

For x = 5:
y = (5 - 5)^2 - 1
y = (0)^2 - 1
y = -1

For x = 6:
y = (6 - 5)^2 - 1
y = (1)^2 - 1
y = 0

Step 3: Write the ordered pairs (x, y) based on the calculated y values.

For x = 4, the ordered pair is (4, 0)
For x = 5, the ordered pair is (5, -1)
For x = 6, the ordered pair is (6, 0)

In the given range, there are 3 ordered pairs (x, y) that satisfy the equation y=(x-5)^2-1.

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In seven years, the student population at the Brentwood Student Center will be approximately 4,174.

Using the given terms, the current student population at the Brentwood Student Center is 2,500 and the enrollment increases at a rate of 6% each year. To find the student population closest to seven years from now, we'll use the formula for exponential growth:

Future Population = Current Population × (1 + Growth Rate)^Number of Years

In this case, the future population will be:

Future Population = 2,500 × (1 + 0.06)^7

After calculating, we get:

Future Population ≈ 4,174

So, to the nearest whole number, the student population at the Brentwood Student Center will be approximately 4,174  in seven years.

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70 percent of the guests at this hotel used the valet service.

To find the percentage of guests who used the valet service, we can follow these steps:

1. Add the number of guests who used the valet service (280) and those who used the public parking garage (120) to find the total number of guests with cars: 280 + 120 = 400 guests.

2. Divide the number of guests who used the valet service (280) by the total number of guests with cars (400).

3. Multiply the result by 100 to convert it into a percentage.

So, let's calculate the percentage:

(280 / 400) * 100 = 0.7 * 100 = 70%

Thus, 70% of the guests at this hotel used the valet service.

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The mean of the given stem and leaf plot is 24.5 and the median of the data is 25.

The stem and leaf plot represents the given data as:

| 2 | 4, 5, 6, 9

| 3 | 1, 4, 5, 5, 7, 8

| 4 | 2, 5, 7, 8, 9

To find the mean, we need to add up all the values and divide by the total number of values.

Mean = (24 + 25 + 26 + 29 + 31 + 34 + 35 + 35 + 37 + 38 + 42 + 45 + 47 + 48 + 49) / 15

= 367 / 15

= 24.5

To find the median, we need to arrange the data in order and find the middle value. As there are 15 data points, the median will be the average of the 8th and 9th data points.

Data in order: 24, 25, 25, 26, 29, 31, 34, 35, 35, 37, 38, 42, 45, 47, 48

Median = (35 + 37) / 2

= 36.

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The value of x in the given circle is 18.1 units.

Given is a circle, where two radii are given one chord is given,

We need to find the value of the x which is also the radius,

We know all the radii in a circle are equal,

So, here the radius = 7.9+10.2 = 18.1 units.

Hence the value of x in the given circle is 18.1 units.

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Answer: The amount of money you would have in 5 years if you could save your entire salary with a 1.2% increase every year would be $87,357.41.

Explanation:

The initial term, a₁, is $74,000, and the common ratio, r, is 1 + 1.2% = 1.012. To find the sum of the first 5 terms, we use the formula for a finite geometric series:

S₅ = a₁(1 - r⁶)/(1 - r)

Plugging in the values, we get:

S₅ = $74,000(1 - 1.012⁵)/(1 - 1.012) = $87,357.41 (rounded to the nearest cent)

Therefore, if you save your entire salary, you would have approximately $87,357.41 in 5 years with a 1.2% increase every year.

To express the number 2 as a logarithm with a base of 4, you would write it as log₄(16). This is because 4² = 16.

In general, the logarithm function is the inverse of exponentiation. When we write logₐ(b) = c, it means that a raised to the power of c equals b.

In your example, you want to find the logarithm of 2 with a base of 4, which means you are looking for the exponent to which 4 must be raised to obtain 2.

So, log₄(2) represents the exponent c such that 4 raised to the power of c equals 2.

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Answer:

Step-by-step explanation:

Answer:

13 years

Step-by-step explanation:

Intuition for how sons can collectively win after a certain period of time:- After a certain period of time the father's age will increase by that certain period of time (say 5 years) but for the sons (since there are 3 of them) their collective age will increase by three times that of their father (5 for each 1 one them). Therefore there exist a time after which collective increase in sons' age can cover the current gap of 13 years.

Answer:

18 cent

Step-by-step explanation:

The  true triangle  statement regarding the diagram are:

1.  m∠5 + m∠6 = 180°  ________Linear Pair

2. ∠ 2+ ∠ 3 = ∠ 6________Exterior angle Property of Triangle

3. m∠2 + m∠3 + m∠5 = 180°________Triangle Sum Property

From the question,  Δ ABC with Exterior angles as ∠ 1 , ∠ 4 ,and ∠ 6

Note that the  Exterior angle Property of Triangle state that An exterior angle of a triangle is equal to the sum of the opposite interior angles.

Hence: For Exterior ∠ 1 :

∠ 1 = ∠ 5 + ∠ 3 ________Exterior angle Property of Triangle

Also,

For Exterior ∠ 4:

∠ 4 = ∠ 5 + ∠ 2 ________Exterior angle Property of Triangle

Also,

In regards to Exterior ∠ 6:

∠ 6 = ∠ 2 + ∠ 3 ________ Exterior angle Property of Triangle

Using Triangle Sum Property, it state that In a triangle sum of the measures of angles is equal to 180° Hence:  m∠2 + m∠3 + m∠5 = 180° ________Triangle Sum Property

The Linear Pair will be: The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees.

Therefore,  m∠5 + m∠6 = 180°  ________Linear Pair

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See full question below

A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options.

m∠5 + m∠3 = m∠4

m∠3 + m∠4 + m∠5 = 180°

m∠5 + m∠6 =180°

m∠2 + m∠3 =

m∠6 m∠2 + m∠3 + m∠5 = 180°

The sample space of the number of outcomes is A = 12

Given data ,

To create a tree diagram and list the sample space, we need to consider the possible outcomes at each stage of the event.

First, we have three options for the spinner: Red, Blue, and Yellow.

Now, let's consider the possible outcomes when a letter is chosen from the word MATH

The total number of outcomes A = 12 outcomes

where A = { RM , BM , YM , RA , BA , YA , RT , BT , YT , RH , BH , YH }

       Red             Blue            Yellow

       / \             / \             / \

      M   A           M   A           M   A

     /     \         /     \         /     \

    T       H       T       H       T       H

Hence , the number of outcomes is A = 12 and the tree diagram is solved

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Sean's percentage profit is 20% on selling 24 chocolate bars.

Sean's cost price for each chocolate bar is:

£10 / 24 bars = £0.4167 per bar

Sean sells each chocolate bar for 50p, which is £0.5

Sean's revenue from selling all 24 chocolate bars is:

24 bars x £0.5 per bar = £12

Sean's profit is the difference between his revenue and his cost:

Profit = £12 - £10 = £2

To calculate the percentage profit, we can use the following formula:

Percentage profit = (Profit / Cost price) x 100%

So, plugging in the values we get:

Percentage profit =[tex](2 / 10) x 100% = 20%[/tex]= 20

Therefore, Sean's percentage profit is 20%. He earned a profit of £2 on his initial investment of £10, which is equivalent to a 20% return on investment.

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The correct statement is given as follows:

The function g(t) reveals the market value of the house increases by 3.6% each year.

An exponential function has the definition presented as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

The parameter b for this problem is given as follows:

b = 1.036.

As the parameter b has an absolute value greater than 1, the function is increasing, with a rate given as follows:

1.036 - 1 = 0.036 = 3.6% a year.

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To do this, we first need to factor the denominator of the integrand into linear factors. In this case, the denominator is given as 24 + 16.02 = 40, which is already a factorization. Therefore, we can write:

∫ (6x3 + 10x2 + 64x + 96)/(24 + 16.02) dx = ∫ [(a/(24 + 16.02)) + (b/(22 + 16))] dx

where a, b are constants that we need to find. To do this, we can use the method of partial fractions, which involves equating the coefficients of like terms on both sides of the equation. Specifically, we can write:

6x3 + 10x2 + 64x + 96 = (a/(24 + 16.02))(22 + 16) + (b/(22 + 16))(24 + 16.02)

Multiplying both sides by the common denominator (24 + 16.02)(22 + 16), we get:

(6x3 + 10x2 + 64x + 96)(24 + 16.02)(22 + 16) = a(22 + 16) + b(24 + 16.02)(24 + 16)

Expanding both sides and collecting like terms, we get a system of two linear equations in two unknowns:

(24 + 16.02)(22 + 16)a + (24 + 16.02)(24 + 16)b = 6(24 + 16.02)(22 + 16) + 10(22 + 16)(24 + 16.02) + 64(24 + 16.02) + 96(22 + 16)

(22 + 16)a + (24 + 16.02)b = 6(22 + 16) + 10(24 + 16.02) + 64 + 96

Solving this system (which involves some algebraic manipulation) gives:

a = -6/5
b = 18/5

Therefore, we can write:

∫ (6x3 + 10x2 + 64x + 96)/(24 + 16.02) dx = (-6/5) ∫ (22 + 16)/(24 + 16.02) dx + (18/5) ∫ (24 + 16.02)/(22 + 16) dx

To evaluate these integrals, we can use the substitution u = 24 + 16.02 in the first integral and u = 22 + 16 in the second integral. This gives:

∫ (6x3 + 10x2 + 64x + 96)/(24 + 16.02) dx = (-6/5) ln|24 + 16.02| + (18/5) ln|22 + 16| + C

where C is the constant of integration. Finally, using the given expression for the integral, we can equate coefficients of like terms to obtain:

16x3 + 10x2 + 64x + 96 = (6/5)(24 + 16.02) ln|24 + 16.02| - (18/5)(22 + 16) ln|22 + 16| + C

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The sets (2,6) and [1,5] are not the same mathematically  but they do have some overlap.

The first pair, (2,6), signifies a number line interval that is open and commences at 2, concluding at 6, while excluding the endpoints.

So one can say that the closed interval on the number line between 1 and 5, including both endpoints, is represented by the set [1,5]. any integer that is seen between 1 and 5, inclusive, is included in this set.

Although there is some similarity between the two groups, namely the presence of numbers 2 to 5, they are distinct from each other. The numerical interval (2,6) does not contain the values 2 and 6, whereas those two numbers are part of the range [1,5].

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See transcribed text below

Mathematically these two sets are the same or are they a way of rewriting the other or are they different? (2,6) and [1,5]

Answer:

x = 16

Step-by-step explanation:

(2x + 16) = 48

Subtract 16 with the positive 16 to cancel the numbers.

Subtract 16 with 48.

2x = 32

divide 32 by 2 to isolate the x.

32/2 = 16

x = 16

Answer:

c/π=d

explanation:

d × π = c

divide c to isolate d

Answer: I would multiply pie by a diameter until it equals 16.

(I know this probably isn’t the professional way but it should work.

The length of the rectangle is 15 cm and the width is 9 cm.

Let's start by setting up the equations we need to solve:

L = 2W - 3 (the length is 3 cm less than twice the width)

2L + 2W = 48 (the perimeter is 2 times the length plus 2 times the width)

Now we can substitute the first equation into the second equation and solve for W:

2(2W - 3) + 2W = 48

4W - 6 + 2W = 48

6W = 54

W = 9

Now that we know the width is 9 cm, we can substitute this value back into the first equation and solve for L:

L = 2(9) - 3

L = 15

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Answer:

Step-by-step explanation:

(a)

x + y = -1

3x = 4 - 3y

from 1st equation we can write x = -1 - y and put this x in the 2nd equation

3(-1 -y) = 4 -3y

-3 -3y = 4 -3y

now here y is getting cancel so that means this two equation has no common solution.

(b)

3x - 4y +2 = 0

10 -10y = 10y -15x

from 1st equation we can write x = (4y - 2)/3 and put this x in the 2nd equation

10 = 10y +10y -15((4y - 2)/3)

2 = 2y +2y - 3((4y -2)/3)

2 = 4y - (4y - 2)

again here 4y and 4y getting cancel so both the equation has no common solution.

Total outcomes when we spin the spinner and flip a coin = 12

In the figure

In spinner the labelled number are from 1 to 6

And for a coin there are two outcomes head and tail

Therefore,

total number of outcomes for spinner = 6

total number of outcomes for a coin   = 2

Then the number of outcomes when both are performed once

= 6x2

= 12

Hence total outcomes = 12

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When both the increasing acreage and the increasing yield are considered, the total number of bushels of corn currently increasing at a rate of 4000 bushels per year.

The wheat farmer is currently farming 400 acres of corn and increasing that number by 20 acres per year. He harvests 120 bushels of corn per acre, with an increasing yield of 4 bushels per acre per year.

To determine the rate of increase in the total number of bushels, we need to consider both the increasing acreage and the increasing yield.

First, let's find the increase in bushels due to the increasing acreage:
20 acres/year * 120 bushels/acre = 2400 bushels/year

Next, let's find the increase in bushels due to the increasing yield:
400 acres * 4 bushels/acre/year = 1600 bushels/year

Now, add both increases together to find the total increase in bushels:
2400 bushels/year + 1600 bushels/year = 4000 bushels/year

So, the total number of bushels of corn is currently increasing at a rate of 4000 bushels per year.

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The situation represents exponential decay.

The rate of growth or decay, r, is equal to 0.02.

So the depth of the lake each year is 0.98 times the depth in the previous year.

It will take between 11 and 12 years for the depth of the lake to reach 26. 7 meters.

The situation represents exponential decay, as the depth of the lake decreases by a constant percentage each year. The rate of decay is 2% per year, so the rate of growth or decay, r, is equal to 0.98 (1 - 0.02). This means that the depth of the lake each year is 0.98 times the depth in the previous year.

To find the number of years it will take for the depth of the lake to reach 26.7 meters, we can use the formula for exponential decay:\

D = D₀ *[tex]e^{(-rt)[/tex]

where D is the current depth, D₀ is the initial depth, r is the rate of decay, and t is the number of years.

Substituting the given values, we get:

26.7 = 30 * [tex]e^{(-0.02t)[/tex]

Solving for t, we get:

t = ln(26.7/30) / (-0.02) ≈ 11.33

Therefore, it will take between 11 and 12 years for the depth of the lake to reach 26.7 meters.

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The correct equation is,

⇒ (5 - 6i) (2 + 4i)

Let us assume that;

⇒ (a + bi) (c + di)

⇒ ac + adi + bci + bdi²

⇒ (ac − bd) + (ad + bc)i

Matching coefficients:

30 < ac − bd < 80

ad + bc = 0

Hence, We need to pick four integers between -9 and 9 such that these two equations are satisfied.  One possible combination is:

a = 8, b = -4, c = 6, d = 3

The number would be:

ac − bd = (8)(6) − (-4)(3) = 60

Puzzle 2

Using the result from Puzzle 1:

ac − bd = 34

ad + bc = 8

Like before, it makes sense to assume b is negative.  With some trial and error, one possible answer is:

a = 5, b = -6, c = 2, d = 4

Thus, The correct equation is,

⇒ (5 - 6i) (2 + 4i)

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Let's use "a" to represent the length of the equal sides of the isosceles triangle, and let's use "b" to represent the length of the third side. We're told that one of the equal sides is 2x + 1ft long, so we can set up an equation:

2a + b = 55

We're also told that the other two sides are both 3x - 14ft long, so we can set up another equation:

a = 3x - 14

Now, we can substitute the second equation into the first equation and solve for "b":

2a + b = 55

2(3x-14) + b = 55

6x - 28 + b = 55

b = 83 - 6x

Now, we can substitute both equations into the equation a = 3x - 14 and solve for "x":

3x - 14 = 2x + 1 + 3x - 14

6x - 27 = 0

x = 4.5

Finally, we can substitute "x" into our equations to find the lengths of the sides:

a = 3x - 14 = 3(4.5) - 14 = 0.5

b = 83 - 6x = 83 - 6(4.5) = 55

So the length of the equal sides is 0.5ft, and the length of the third side is 55ft. Therefore, the lengths of the sides of the isosceles triangle are 0.5ft, 0.5ft, and 55ft.

To find out how much lace Ms. Regan needs to purchase, we first need to calculate the circumference of the circular quilt. We know that the area of the quilt is 28.26 square feet, and we can use the formula A = πr^2 to find the radius of the quilt.

28.26 = 3.14 x r^2

r^2 = 9

r = 3

Now that we know the radius is 3 feet, we can use the formula C = 2πr to find the circumference of the quilt.

C = 2 x 3.14 x 3

C = 18.84 feet

Therefore, Ms. Regan needs to purchase 18.84 feet of lace to go around the outside of her circular quilt.

In summary, to find out how much lace Ms. Regan needs to purchase, we need to calculate the circumference of the circular quilt. We do this by first finding the radius using the formula A = πr^2. Once we know the radius, we can use the formula C = 2πr to find the circumference. In this case, the circumference is 18.84 feet, so Ms. Regan needs to purchase that amount of lace.

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The point(s) on the curve y = x²/6 closest to the point (0,0) are (0,0) and (±√2, 2/3).

To find the point(s) on the curve y = x²/6 closest to the point (0,0), we can use the distance formula between two points:

d = √((x₁ - x₂)² + (y₁ - y₂)²)

where (x₁, y₁) is a point on the curve and (x₂, y₂) is the point (0,0).

We want to minimize the distance d, which is equivalent to minimizing d². Therefore, we can minimize:

d² = (x₁ - 0)² + (y₁ - 0)²

= x₁² + y₁²

subject to the constraint that the point (x₁, y₁) is on the curve y = x²/6.

Substituting y = x²/6 into the expression for d², we get:

d² = x₁² + (x₁²/6)

= (7/6)x₁²

To minimize d², we minimize x₁². Since x₁² is always non-negative, the minimum occurs when x₁² = 0 or when the derivative of d² with respect to x₁ is zero.

Taking the derivative of d² with respect to x₁, we get:

d²/dx₁ = (7/3)x₁

Setting this equal to zero, we get x₁ = 0.

Therefore, the point (0,0) is one of the closest points on the curve to the point (0,0).

To find the other closest point(s), we can solve y = x²/6 for x² and substitute it into the expression for d²:

x² = 6y

d² = 7x²/6 = 7y

Therefore, to minimize d², we need to minimize y. Since y is always non-negative, the minimum occurs when y = 0 or when the derivative of d² with respect to y is zero.

Taking the derivative of d² with respect to y, we get:

d²/dy = 7

Setting this equal to zero, we get y = 0.

Substituting y = 0 into y = x²/6, we get x = 0. Therefore, the point (0,0) is one of the closest points on the curve to the point (0,0).

To find the other closest point, we can solve y = x²/6 for x:

x² = 6y

x = ±√(6y)

Substituting this into the equation for y, we get:

y = (√(6y))²/6 = 2/3

Therefore, the other closest points are (±√2, 2/3).

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Answer:

C. The product of two irrational numbers is irrational.

Example: √3•√3=3