8. Students were asked to simplify the following expressions on a test. (154 Four different student answers are shown below: Student A 21 - 3h 15-4h+6+h Student B 18h Student C 3h + 21 600p Student D 21 +- 3h Which student or students simplified the expression correctly? Explain​

The solution to the logarithmic equation [tex]2log12(-8x) = 6[/tex] is [tex]x = -9/32[/tex] .

What are logarithmic properties ?

Logarithmic properties are the rules that govern the behavior of logarithmic functions. These properties are important in simplifying logarithmic expressions and solving logarithmic equations. Some of the commonly used logarithmic properties include:

Product property: [tex]logb(xy) = logb(x) + logb(y)[/tex]

This property allows us to simplify the logarithm of a product of two numbers into the sum of logarithms of the individual numbers.

Quotient property: [tex]logb(x/y) = logb(x) - logb(y)[/tex]

This property allows us to simplify the logarithm of a quotient of two numbers into the difference of logarithms of the individual numbers.

Power property:[tex]logb(x^y) = ylogb(x)[/tex]

This property allows us to simplify the logarithm of a power of a number by bringing the exponent outside of the logarithm and multiplying it with the logarithm of the base.

Change of base formula: [tex]logb(x) = logc(x) / logc(b)[/tex]

This property allows us to change the base of a logarithm by dividing the logarithm of the number by the logarithm of the base in a different base.

Solving the given logarithmic equation :

The equation can be solved by using logarithmic properties and basic algebraic manipulation.

We can begin by using the property that states [tex]loga(b^n) = nloga(b)[/tex] for any base a and any positive real number b. Applying this property, we can rewrite the left side of the equation as:

[tex]log12((-8x)^2) = log12(64x^2)[/tex]

Next, we can use the property that states [tex]loga(b) = c[/tex] is equivalent to [tex]a^c = b[/tex]. Applying this property, we can rewrite the equation as:

[tex]12^{2log12(64x^2)} = 12^6[/tex]

Simplifying the left side, we get:

[tex]64x^2 = 12^6 / 12^2[/tex]

[tex]64x^2 = 144[/tex]

Dividing both sides by 64, we get:

[tex]x^2 = 144/64[/tex]

[tex]x^2 = 9/4[/tex]

Taking the square root of both sides, we get:

[tex]x=\pm 3/2[/tex]

However, we need to check the solutions for extraneous roots since the original equation has a logarithm with a negative argument. We can see that the solution x = 3/2 is extraneous since it results in a negative argument for the logarithm. Therefore, the only valid solution is x = -9/32.

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The scale factor of the following pair of similar polygons after the dilation is 0.7

Given

The pair of similar polygons

From the pair of similar polygons, we have the following corresponding side lengths

Pre-image of the polygon = 30

Image of the polygon = 21

The scale factor of the similar polygons is then calculated as

Scale factor = 21/30

Evaluate the quotient

Scale factor = 0.7

Hence, the scale factor is 0.7

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In the fractions prompts given, the correct output are:

A fraction represents a part of a whole. It consists of a numerator and a denominator, with the numerator indicating the number of parts and the denominator indicating the total number of parts.

The calculations are given as follows;

1 )

= (1/2) x (1/4) [dividing by a number is the same as multiplying by its reciprocal]

= 1/8

2) (1/5) ÷ 2

= (1/5) x (1/2)

= 1/10

3)

(1/3) ÷ 5

= (1/3) x (1/5)

= 1/15

4) (1/4) ÷ 4

= (1/4) x (1/4)

= 1/16

5) One day, Amy baked a cake and wanted to divide it equally among 3 of her friends. She realized she only had half a cake left, so she decided to divide it into equal parts. Each friend received 1/6 of a cake. To check her calculation, she multiplied 1/6 by 3 and got 1/2. Thus, (1/2) ÷ 3 = 1/6, since dividing by 3 is the same as multiplying by 1/3.

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Answer:

25 computers per hour

Step-by-step explanation:

look ate the point 2 hours corresponding to 50 computers

50/2 = 25/1 or 25 computers per hour

ANSWER

Jules paid $0.70 for one pencil

STEPS

Cost of One Pencil.

Let's assume the cost of one notebook be x and one pencil be y.

According to the given information,

Shanika bought 1 notebook and 5 pencils at a cost of $10.00. Therefore,

1x + 5y = 10 --- equation 1

Jules paid $14.90 for 1 Notebook and 12 pencils. Therefore,

1x + 12y = 14.9 --- equation 2

We need to find the cost of one pencil, i.e., y.

We can solve the above equations simultaneously to find the value of y.

Multiplying equation 1 by 12 and equation 2 by 5, we get:

12x + 60y = 120 --- equation 3

5x + 60y = 74.5 --- equation 4

Subtracting equation 4 from equation 3, we get:

7x = 45.5

x = 6.5

Substituting the value of x in equation 1, we get:

1(6.5) + 5y = 10

5y = 3.5

y = 0.7

Therefore, Jules paid $0.70 for one pencil.

ChatGPT

Answer:

Jules pay:

$0.7

Step-by-step explanation:

1n + 5p = 10                  Eq. 1

1n + 12p = 14.9              Eq. 2

n = cost of one notebook

p = cost of one pencil

From Eq. 1

n = 10 - 5p               Eq. 3

From Eq. 2

n = 14.9 - 12p            Eq. 4

Equalizing Eq. 3 and Eq. 4:

10 - 5p = 14.9 - 12p

12p - 5p = 14.9 - 10

7p = 4.9

p = 4.9 / 7

p = 0.7

From Eq. 3:

n = 10 - 5p

n = 10 - 5*0.7

n = 10 - 3.5

n = 6.5

Check:

From Eq. 2:

n + 12p = 14.9

6.5 + 12*0.7 = 14.9

6.5 + 8.4 = 14.9

Then:

1 pencil = $0.7

Answer:

C is your answer

Step-by-step explanation:

in my opinion, i think it would be the mode.

Answer:

2.31 Years

Step-by-step explanation:

To calculate the time it will take for ₹5000 to grow to ₹5618 with a 6% annual interest rate when compounded annually, we can use the following formula:

A = P(1 + r/n)^(nt)

Where:

A = the final amount (₹5618)

P = the principal amount (₹5000)

r = the annual interest rate (6% or 0.06)

n = the number of times the interest is compounded per year (1, since it's compounded annually)

t = the time period in years

Plugging in the values, we get:

5618 = 5000(1 + 0.06/1)^(1t)

Simplifying:

1.1236 = 1.06^t

Taking the natural logarithm of both sides:

ln(1.1236) = ln(1.06^t)

Using the power rule of logarithms:

ln(1.1236) = t ln(1.06)

Solving for t:

t = ln(1.1236) / ln(1.06)

t ≈ 2.31 years

Therefore, it will take approximately 2.31 years for ₹5000 to grow to ₹5618 at a 6% annual interest rate when compounded annually.

The function is constant from approximately x = -4 to x = -3 and from approximately x = -1 to x = 1. So, the constant intervals are (-4, -3) and (-1, 1)

A function, which produces one output from a single input, is an illustration of a rule. The picture was obtained from Alex Federspiel. The equation y=x2 serves as an example of this.

a) We can see that the function is increasing from approximately x = -3 to x = -1 and from approximately x = 1 to x = 2.5. So, the increasing intervals are (-3,-1) and (1, 2.5)

(b) We can see that the function is decreasing from approximately x = -2 to x = -0.5 and from approximately x = 3 to x = 4. So, the decreasing intervals are (-2, -0.5) and (3, 4)

(c) We can see that the function is constant from approximately x = -4 to x = -3 and from approximately x = -1 to x = 1. So, the constant intervals are (-4, -3) and (-1, 1)

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10/73 is the value of x in inequality.

What does the word "inequality" mean?

In mathematics, inequalities describe the connection between two values that are not equal. Equal does not imply inequality. The "not equal symbol ()" is typically used to indicate that two values are not equal.

                    However different inequalities are used to compare the values to determine if they are less than or higher than. The term "inequality" refers to a relationship between two expressions or values that is not equal to one another. Inequality originates from an imbalance, thus.

the inequality  12≥ 73x + 2

                    = 12 - 2  ≥  73x

                     = 10 ≥ 73x

                   = 10/73 ≥ x

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As a result, y has a mean of 203 and a variance of 85 as let x1 and x2 be two independent random variables both with mean 10 and variance 5.

A variable is a symbol or letter that is used to indicate a variable quantity in mathematics. The context or issue under consideration can alter the value of a variable. In order to express relationships between quantities, variables are frequently utilized in equations, formulae, and functions. For instance, x and y are variables in the equation y = mx + b, which depicts the linear relationship between x and y. Variables in statistics can reflect various traits or features of a population or sample, such as age, body mass index, or income.

given

To get the mean and variance of y, we can apply the characteristics of expected value and variance:

We can start by determining the expected value of y:

E[y] = 2E[x1] = E[2x1x2 + 3x1 + 2]

By the linearity of expectation, E[x2] + 3E[x1] + 2 is 2(10)(10) + 3(10) + 2 = 203.

Next, we may determine y's variance:

Var(y) = Var(3x1 + 2 + 2 + 3x1 ) = 4

Var(x1)

Var(x2) + 9

Var(x1) + Var(constant) = 4(5)(5) + 9(5) + 0 = 85 since x1 and x2 are independent.

As a result, y has a mean of 203 and a variance of 85 as let x1 and x2 be two independent random variables both with mean 10 and variance 5.

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Answer: 61

Step-by-step explanation:

explination in image

We have that, using Euclid's algorithm, we find the inverse of 200 modules 1001 is -5 (or 1001+5).

To find the inverse of a module m using Euclid's algorithm, the steps are as follows:

1. Calculate the greatest common divisor (GCD) of a and m using the Euclidean algorithm.

2. Let a = GCD * s + m*t, where s is the inverse of a module m.

3. The GCD in terms of a and my is written as 1 = m-s*a.

4. Find s = -a, so the inverse of a module m is -a (or m+s).

For example, a = 2, m=17, so GCD = 1 = 17-8*2 and the inverse of 2 modulo 17 is -8 (or 17+8). Similarly, for a = 34, m= 89, the GCD = 1 = 89-34*2 and the inverse of 34 modulo 89 is -34 (or 89+34). Finally, for a = 200, m= 1001, the GCD = 1 = 1001-5*200 and the inverse of 200 modulo 1001 is -5 (or 1001+5).

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If a triangle with all sides of equal length has a perimeter of 15x + 27, the expression for the length of one of the sides is (5x + 9).

The perimeter of a triangle is the sum of the lengths of all three sides. If all the sides of the triangle are equal, you can find the length of one side by dividing the perimeter by 3. Here, the perimeter is given as 15x + 27. Therefore, the length of one side will be (15x + 27) / 3 = 5x + 9. Hence, an expression for the length of one of the sides is (5x + 9).

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The volume of lemon juice in the glass is 0.38 cubic inches.

Explanation:

Given,

Let the volume of the lemonade in the glass be V cubic inches

Therefore, the volume of lemon juice in the lemonade is [tex]$\frac{1}{12}$[/tex] V cubic inches

Volume of water in the lemonade is [tex]$\frac{11}{12}$[/tex] V cubic inches

The volume of the cylindrical glass is given by:

[tex]$V_{\text{cylindrical glass}} = \pi r^2h$[/tex]

Here,

Radius r = 1 inch

Height h = 6 inches

[tex]$V_{\text{cylindrical glass}} = \pi r^2h = \pi (1)^2(6) = 6 \pi$[/tex]

Since the glass is half full of lemonade, the volume of lemonade in the glass is:

[tex]$V_{\text{lemonade}} = \frac{1}{2}V_{\text{cylindrical glass}} = \frac{1}{2} 6 \pi = 3\pi$[/tex]

The volume of lemon juice in the lemonade is given by:

[tex]$V_{\text{lemon juice}} = \frac{1}{12}V$[/tex]

Therefore

[tex]$V_{\text{lemon juice}} = \frac{1}{12}3\pi = \frac{1}{4}\pi = 0.7854$[/tex] cubic inches

Hence, the volume of lemon juice in the glass is 0.38 cubic inches (rounded to the nearest hundredth).

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Answer:

D. The electrician charges $23 per hour.

Step-by-step explanation:

C(h)= 23h+30 is in the form y=mx +b

$30 is the initial fee (b)

$23 is the amount charged per hour (h)

Answer:

Step-by-step explanation:

lknk

pihpin

87647fpom';

oiy69877t7gbkjb

Answer:

s= 6

Step-by-step explanation:

The speed s (in miles per hour) of a car can be given by s = √(30fg), where fis the

coefficient of friction and d is the stopping distance (in feet). The table shows the

coefficient of friction for different surfaces. You are driving 35 miles per hour on an

icy road when a deer jumps in front of your car. How far away must you begin to

brake to avoid hitting the deer? Round your answer to the nearest whole integer. NO

UNITS NEEDED

The null hypothesis is rejected whenever "there is a low probability that the obtained results could be due to random error." The correct answer is Option B.

The null hypothesis is a statistical hypothesis used to test the difference between two sample data groups. The null hypothesis is the hypothesis that the sample statistics are not significantly different. Any significant differences between the sample data are seen as supporting the alternative hypothesis.

A null hypothesis is often expressed as "no difference," "no correlation," or "no significant effect." For example, the null hypothesis for an experiment comparing two groups of people could be "there is no difference between the two groups." When the null hypothesis is rejected, it means that the results of the experiment are statistically significant, and the alternative hypothesis is supported.

Therefore, the null hypothesis is rejected whenever there is a low probability that the obtained results could be due to random error.

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Answer: false

Step-by-step explanation:

The matrix of predicted rankings UVT is [[48,24,12],[0,0,0],[24,12,6]].

The matrix of predicted rankings UVT can be calculated using the formula UVT = UV.The provided initial values for U() and V(0) are as follows:U() = [[2,1],[1,0],[3,-1]]V(0) = [[24,12,6],[0,0,0],[12,6,3]]Using the above values, the matrix of predicted rankings UVT can be computed as follows:UVT = UVU = [[2,1],[1,0],[3,-1]]V = [[24,12,6],[0,0,0],[12,6,3]]UVT = [[2,1],[1,0],[3,-1]] x [[24,12,6],[0,0,0],[12,6,3]]= [[48,24,12],[0,0,0],[24,12,6]]Therefore, the matrix of predicted rankings UVT is [[48,24,12],[0,0,0],[24,12,6]].

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Answer:

x=1

Step-by-step explanation:

2.5+x=3.5

3.5-2.5=x

1=x

x=1

The x ≤ 32Putting x = 24 in the expression we get RC = 96Therefore, the value of RC is 96.

In order to find RC, we will make use of the given information in the following manner: Given that ABDC is a parallelogram. Hence, AB = DC. We have also been given that PQ = 735 and QR = 112.Now, extend PQ to meet DC at S.Let PS = x; then DS = DC - x = AB - x. (Since AB = DC)We have that PS/SP = QR/RB (Since PQR is similar to DBR)Therefore, we getx/SP = 112/(AB - x)We can cross multiply and simplify to getSP = (112* x)/(AB - x)......(1)Further, we have that AQ/QB = SP/BR (Since PQR is similar to AQB)Therefore, we getx/(AB - x) = SP/BROn substituting the value of SP from equation (1) above, we getx/(AB - x) = (112* x)/(BR*(AB - x))Therefore, we getBR = (x*(AB - x)*QR)/[PQ*(AB - 2*x)]BR = (x*(32 - x)*112)/(735*(32 - 2*x))BR = (56*x*(16 - x))/(245*(16 - x))BR = (56*x/245)Since the sum of all sides of a parallelogram is equal to the sum of its opposite sides, we have thatRC + QR = AB + AQ - QBRearranging the terms we getRC = AB + AQ - QB - QR......(2)Now, AQ = PQ - APSubstituting the values of PQ and AP we get AQ = 735 - (DC - x) = 735 - 32 + x = x + 703Also, QB = AB - AQ = (32 - x) - (x + 703) = -x - 671Substituting the values of AQ and QB in equation (2) above we getRC = 32 + (x + 703) + (x + 671) - 112RC = 48 + 2xRC = 2(x + 24)We know that AB = 32, hence, x ≤ 32Putting x = 24 in the expression we get RC = 96Therefore, the value of RC is 96.

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Answer: I couldn't honestly help with that I would if I could

Step-by-step explanation:

Answer:

Step-by-step explanation:

Given f(x) = 2x³ +x² +ax +b has a factor (x -2) and a remainder of 18 when divided by (x -1), you want to know a, b, and the factored form of f(x).

If (x -2) is a factor, then the value of f(2) is zero:

  f(2) = 2·2³ +2² +2a +b = 0

  2a +b = -20 . . . . . . . subtract 20

If the remainder from division by (x +1) is 18, then f(-1) is 18:

  f(-1) = 2·(-1)³ +(-1)² +a·(-1) +b = 18

  -a +b = 19 . . . . . . . . . . add 1

Subtracting the second equation from the first gives ...

  (2a +b) -(-a +b) = (-20) -(19)

  3a = -39

  a = -13

  b = 19 +a = 6

The values of 'a' and 'b' are -13 and 6, respectively.

We can find the quadratic factor using synthetic division, given one root is x=2. The tableau for that is ...

  [tex]\begin{array}{c|cccc}2&2&1&-13&6\\&&4&10&-6\\\cline{1-5}&2&5&-3&0\end{array}[/tex]

The remainder is 0, as expected, and the quadratic factor of f(x) is 2x² +5x -3. Now, we know f(x) = (x -2)(2x² +5x -3).

To factor the quadratic, we need to find factors of (2)(-3) = -6 that have a sum of 5. Those would be 6 and -1. This lets us factor the quadratic as ...

  2x² +5x -3 = (2x +6)(2x -1)/2 = (x +3)(2x -1)

The factored form of f(x) is ...

  f(x) = (2x -1)(x -2)(x +3)

Answer:

Step-by-step explanation:

To find the possible length and breadth of the rectangle, we need to factor the given expression:

x^2 - 6x + 8 = (x - 4)(x - 2)

Therefore, the length and breadth of the rectangle can be any combination of (x-4) and (x-2).

For example, if we choose (x-4) as the length and (x-2) as the breadth, we have:

Length = x - 4

Breadth = x - 2

Conversely, if we choose (x-2) as the length and (x-4) as the breadth, we have:

Length = x - 2

Breadth = x - 4

So, the possible length and breadth of the rectangle are (x-4) and (x-2), and vice versa.

Answer:(x-4) and (x-2)

Step-by-step explanation:

The solution to the equation √(1 + 25/144) = 1 + x/12 is x = 5/6. This was achieved by simplifying the left side of the equation and isolating x on one side.

To solve the equation √(1 + 25/144) = 1 + x/12, we start by simplifying the left side of the equation. The expression inside the square root can be simplified to (144 + 25)/144 = 169/144. Taking the square root of this fraction gives us √(169/144) = (13/12).

Next, we subtract 1 from both sides of the equation to isolate x on one side: (13/12) - 1 = x/12. This simplifies to 1/12 = x/12.

Finally, we multiply both sides by 12 to solve for x: x = (1/12)*12 = 5/6.

So the solution to the equation √(1 + 25/144) = 1 + x/12 is x = 5/6.

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Answer: It should be 470 cm^2

Step-by-step explanation:

Answer: D. Yes, the sum of the square of the legs is equal to the square of the hypotenuse.

Step-by-step explanation:

Pythagorean theorem is a^2 + b^2 = c^2

given side length of 6 and 8 and hypotenuse of 10

6^2 + 8^2 = 10^2

36 + 64 = 100

100 = 100

so yes, the sum of the square of the legs = the square of the hypotenuse.

Answer:

3 packages

Step-by-step explanation:

61 - 20 * 2 (small grain sandpaper) = $21 remain. The largest grain costs $7, so 21/7 = 3 packages of the largest grain sandpaper was bought.

Hope this helped!

The value of sec A as required to be determined in the task content is; -√13 / 2.

As evident from the task content; it follows that the terminal ray of angle A, drawn in standard position, passes through the point (-4, -6).

Therefore, the length that the line from the origin to A has length;

L = √((-4)² + (-6)²)

L = √52.

On this note, it follows that the value of sec A which is represented by; hypothenuse/ adjacent is;

sec (A) = -√52 / 4

sec (A) = -√13 / 2.

Ultimately, the value of sec (A) as required is; -√13 / 2.

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