Suppose your average, after taking 4 quizzes, is 73(out of 100). What must your average be on the next 5 quizzes to increase your average to 88 out of 100?

There are 22,100 possible hands of 3 cards from a 52-card deck using combinations. The probability of dealing three eights in a row from the deck is approximately 0.000181.

This question involves calculating probabilities and combinations related to card hands from a deck of 52 cards.

Possible Hands of 3 Cards:

To determine the number of possible hands of 3 cards from a 52-card deck, we use the combination formula: C(n, k) = n! / [k! * (n - k)!], where n is the total number of cards and k is the number of cards to choose.

C(52, 3) = 52! / [3! * (52 - 3)!] = 52! / (3! * 49!) = (52 × 51 × 50) / (3 × 2 × 1) = 22100. Therefore, there are 22,100 possible hands of 3 cards.

Probability of Dealing Three Eights:

The probability of dealing three eights in a row can be determined as follows:

Therefore, the probability of dealing three eights is: (4/52) × (3/51) × (2/50) = 24 / 132600 = 1 / 5525 ≈ 0.000181.

This detailed explanation helps in understanding how combinations and probabilities in card games are calculated.

The range of the given function, f(x) = x^2 - 4x + 1, is {y: y ≥ -3}. This is determined by completing the square to find the vertex form of the parabola, which indicates that the minimum y-value is -3 (the y-coordinate of the vertex) and the function values extend to infinity.

The question asks us to find the range of the function f(x) = x^2 - 4x + 1. In mathematics, the range of a function is the set of output values (the y-values) that the function produces when we substitute the domain (the x-values) into the function.

To determine the range of a quadratic function like this, we could complete the square for the function and adjust it to the vertex form of a parabola, which is f(x) = a(x - h)^2 + k, where k is the minimum or maximum y-value of the parabola.

For the function f(x) = x^2 - 4x + 1, the completed square form is f(x) = (x - 2)^2 - 3. So, the vertex of the parabola is (2, -3), and since the coefficient of (x - 2)^2 is positive, the parabola opens upwards. This means that the minimum y-value is -3 and the maximum y-value is infinity. So the range of the function is {y: y ≥ -3}.

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When a variable is eliminated from the equation during the equation solving process, the two possible solutions are:

A single unique solution and No solution:

When a variable is eliminated from an equation during the solving process, it results in an equation with only one variable. The two possible solutions are:

A single unique solution: This means that there is one specific value for the remaining variable that satisfies the equation.

No solution: This means that there is no value for the remaining variable that satisfies the equation, resulting in an inconsistent or contradictory statement.

Let's consider an example equation to illustrate these possibilities:

3x + 2y = 10

6x + 4y = 20

To solve this system of equations, eliminate the variable "x" by multiplying the first equation by 2 and subtracting it from the second equation:

[tex](6x + 4y) - 2(3x + 2y) = 20 - 2(10)\\6x + 4y - 6x - 4y = 20 - 20[/tex]

0 = 0

In this case, the variable "x" has been eliminated, and we are left with the equation 0 = 0. This equation is true for all values of "y".

Therefore, the system of equations has infinitely many solutions, and any value of "y" would satisfy the equations.

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The probability of drawing a red card first followed by a blue card from a hat with five different colored cards without replacement is 0.05 or 5%.

The question asks for the probability of drawing a red card first and a blue card second from a hat containing five different colored cards without replacement. To calculate this, we use the formula for conditional probability and the multiplication rule for independent events. Since the cards are drawn without replacement, the events are dependent.

The probability of drawing the red card first is 1/5, since there is one red card out of five. After drawing the red card, there are four cards left. The probability of drawing the blue card second is thus 1/4, as there is only one blue card left among the four remaining cards. The probability of both events occurring in sequence is the product of the probabilities of each event:

P(Red first and Blue second) = P(Red first) × P(Blue second | Red first) = (1/5) × (1/4) = 1/20 or 0.05.

Therefore, the probability that the first card is red and the second card is blue is 0.05 or 5%.

The slope m of the straight line through the points (6, 5) and (7, 2) is -3.

Slope or the gradient is the number or the ratio which determines the direction or the steepness of the line.

The slope of the line is defined as the ratio of the rise to the run.

To find the slope, we can use the formula:

m = (y₂ - y₁) / (x₂ - x₁)

where (x₁, y₁) = (6, 5) and (x₂, y₂) = (7, 2).

Substituting these values, we get:

m = (2 - 5) / (7 - 6) = -3 / 1 = -3

Therefore, the slope of the straight line through the given pair of points is -3.

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The first battery voltage is 12.

The second battery voltage is 6.

The third battery voltage is 36.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

First battery voltage = M

Second battery voltage = N

Third battery voltage = O

Now,

We can make an expression as,

M + N + 0 = 54 _____(A)

And,

M = 2N  ____(1)

M = (1/3)O ____(2)

Now,

From (1) and (2)

N = M/2

O= 3M

Substituting in (A)

M + N + O = 54

M + M/2 + 3M = 54

4M + M/2 = 54

8M + M = 108

9M = 108

M = 12

Now,

N = M/2 = 12/2 = 6

O= 3M = 3 x 12 = 36

Thus,

First battery voltage = 12

Second battery voltage = 6

Third battery voltage = 36

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The sum of the first 10 terms of the series is 23.

An arithmetic progression(AP) is a sequence or series of numbers such that the difference of any two successive members is a constant. The first term is a, the common difference is d, n is number of terms.

For the given situation,

The series is 0.5,0.9,1.3,1.7.....

Here the first term, a = 0.5,

The common difference, d = [tex]0.9-0.5[/tex]

⇒ [tex]d=0.4[/tex]

Number of terms, n = 10

The formula of sum of n terms is

[tex]S_{n}=\frac{n}{2} [2a+(n-1)d][/tex]

On substituting the above values,

⇒ [tex]S_{10}=\frac{10}{2} [2(0.5)+(10-1)(0.4)][/tex]

⇒ [tex]S_{10}=5 [1+9(0.4)][/tex]

⇒ [tex]S_{10}=5 [1+3.6][/tex]

⇒ [tex]S_{10}=5 [4.6][/tex]

⇒ [tex]S_{10}=23[/tex]

Hence we can conclude that the sum of the first 10 terms of the series is 23.

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The equation that represents the translated function for

[tex]\rm f(x) = \dfrac{1}{5}[/tex]   [tex]\rm is\;\; \rm \bold{g(x) = f(x) +4 = 4.2}[/tex].

Given that the function f(x) = 1/5 is translated up 4 units.  

We have to determine the equation that  represents the translated function

The function f(x) = 1/5  as the function is translated.    

Translation is a concept of mathematics . In translation the shape of the object is simply moved linearly from one point to other point by adding some constant values in the previous coordinates. In translation , there  is no dilation shape and size of the object remains the same.

Let us consider that if the function is translated up with 4 units then the new function is [tex]\rm\bold{ g(x)}[/tex]

So according to the language of question we

[tex]\rm g(x) = f(x) +4[/tex]

[tex]\rm g(x) = \dfrac{1}{5} +4 = \dfrac{21}{5} = 4.5[/tex]

[tex]\rm g(x) = 4.5[/tex]

The equation that represents the translated function for

[tex]\rm f(x) = \dfrac{1}{5}[/tex]      [tex]\rm is\;\; g(x) = f(x) +4 = 4.2[/tex]

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It will take you 15 minutes to walk one mile at a speed of 4 miles per hour.

To determine how long it will take you to walk one mile, we can use the formula:

Time = Distance / Speed

Let's plug in the values:

Distance = 1 mile

Speed = 4 miles per hour

Time = 1 mile / 4 miles per hour

Time = 0.25 hours

Since there are 60 minutes in an hour, we can convert the time to minutes:

Time = 0.25 hours × 60 minutes per hour

Time = 15 minutes

Therefore, it will take you 15 minutes to walk one mile at a speed of 4 miles per hour.

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The Inequality that has the graph shown below is [tex]y \ge -\frac{1}{4}x + 1.[/tex] correct choice is [tex]y \ge -\frac{1}{4}x + 1.[/tex]

This can be seen by noting that the line is solid and shaded above the line, which indicates that all points above the line satisfy the inequality.

Graphs are often used to visually represent inequalities in mathematics. Inequalities compare the relative sizes of two expressions, and their graphs help illustrate the solutions to these inequalities on a number line or in a coordinate plane

The other inequalities in the image are not correct because:

[tex]y > -\frac{1}{4}x + 1[/tex]would have the line shaded below the line.

[tex]y < = -\frac{1}{4}x + 1[/tex] would have the line dashed, not solid.

[tex]y < -\frac{1}{4}x + 1[/tex]would have no points above the line satisfying the inequality.

Therefore, the correct inequality is [tex]y \ge -\frac{1}{4}x + 1.[/tex]

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The expression that shows each factor rounded to the nearest whole number is 32.

Rounding some number to a specific value is making its value simpler (therefore losing accuracy), mostly done for better readability or accessibility.

Rounding to some place keeps it accurate on the left side of that place but rounded off like trimmed from the right in terms of exact digits.

Given that Ella wants to estimate the product (3.6)(7.8).

3.6 rounded to give 4

7.8 rounded to give 8

Therefore, the product would be;

4 x 8 = 32

Hence, the expression that shows each factor rounded to the nearest whole number is 32.

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Final answer:

To estimate the product of 3.6 and 7.8 by rounding to the nearest whole number, round 3.6 down to 3 and 7.8 up to 8, making the estimated expression (3)(8).

Explanation:

The student has asked how to estimate the product of 3.6 and 7.8 by rounding each factor to the nearest whole number.

To do this, we look at each factor and determine which whole number it is closest to. For 3.6, we round down to 3 because it is closer to 3 than to 4. For 7.8, we round up to 8 because it is closer to 8 than to 7. Hence, the expression that represents the product of each factor rounded to the nearest whole number is (3)(8).

The present value that will grow to $7,000 if the annual interest rate is 3.5% compounded quarterly for 9 years is $5,187.72

The Breakdown

To find the present value (PV) that will grow to $7,000 if the annual interest rate is 3.5% compounded quarterly for 9 years, you can use the formula for compound interest:

PV = FV / (1 + (r / n))^(n*t

Where:

PV = Present Value

FV = Future Value ($7,000 in this case)

r = Annual interest rate (as a decimal, so 3.5% is 0.035)

n = Number of times the interest is compounded per year (quarterly, so 4 times)

t = Number of years (9 years in this case)

PV = 7000 / (1 + (0.035 / 4))^(4 * 9)

PV = 7000 / (1 + 0.00875)^(36)

PV = 7000 / (1.00875)^36

Now, calculate (1.00875)^36:

(1.00875)^36 ≈ 1.34985880768

Now, divide $7,000 by this value to find the present value:

PV ≈ 7000 / 1.34985880768 ≈ $5,187.72

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Answer:

8 Hours!!

Step-by-step explanation:

[tex]3x^3y ( 6xy + 7x^2 - 2y )[/tex] is the factored form of [tex]18x^4y^2+21x^5y-6x^3y^2[/tex] .

Given, expression [tex]18x^4y^2+21x^5y-6x^3y^2[/tex] .

To factorize the expression take the common terms present in each part of an expression.

Expression : [tex]18x^4y^2+21x^5y-6x^3y^2[/tex]

Highest power of x which is present in each part = x³

Highest power of y which is present in each part = y

HCF of 18, 21 , 6 = 3

Factorizing the expression:

Take 3x³y as common,

Factored form : [tex]3x^3y ( 6xy + 7x^2 - 2y )[/tex] .

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Answer:

The statement is true.

Step-by-step explanation:

Given : Reflecting the graph of y = sin x across the y-axis is the same as reflecting it across the x-axis.

To find : Is this statement true or false?

Solution:

Rules of reflection :

Along across y-axis if we have a point [tex](x,y)\rightarrow(-x,y)[/tex]

So, y=sinx

The reflection across y-axis→ [tex]y=sin(-x) \Rightarrow y = -sinx[/tex].......[1]

Along across x-axis if we have a point [tex](x,y)\rightarrow(x,-y)[/tex]

so, y=sinx

The reflection across x-axis → [tex]-y=sinx\Rightarrow y= -sinx[/tex] .....[2]

Hence, [1] and [2] both of them are equal

Therefore, the statement is true.

Using number sets, it is found that a whole number cannot be irrational, hence option a is correct.

In this problem, any whole number can be represented by a fraction, hence it cannot be irrational and option a is correct.

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Answer:

A. An Irrational Number

Step-by-step explanation:

Irrational numbers are numbers like pi or angles with degrees, while whole numbers are numbers from 0+

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Answer:

Pool needed to be filled = 7623.62 gallons

Step-by-step explanation:

Length of rectangular pool = 24.5 feet

Width of rectangular pool = 13 feet

Depth of rectangular pool = 4 feet

Now, Volume of the pool = Length × Width × Depth

                                          = 24.5 × 13 × 4

                                          = 1274 cubic feet

Now, 1 cubic feet = 7.48 gallons

⇒ 1274 cubic feet = 7.48 × 1274

                              = 9529.52 gallons

So, Volume of the pool = 9529.52 gallons

Now, the pool is to be filled 80%

So, Pool needed to be filled = 9529.52 × 0.80

                                                 = 7623.62 gallons

The number of gallons of water to the nearest tenth, that are needed to fill the pool to 80% capacity is 7623.6 gallons.

First step:

Volume of the pool=Length × Width × Depth

Volume of the pool= 24.5 ft × 13ft × 4ft

Volume of the pool=1274 cubic feet

Second step:

Number of gallons=(7.48 × 1274)×80%

Number of gallons=9529.5 gallons×80%

Number of gallons=7623.6 gallons

Inconclusion the number of gallons of water to the nearest tenth, that are needed to fill the pool to 80% capacity is 7623.6 gallons.

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