Q. 1: Expand and simplify each of the following expression:



6m- 2(4n+5m+1)-2n + 4




12x + 5(-5y-2z+2) – 2(8x+z) + 7



Q. 2: Factorize the following:



8pq + 20qr – 16s




4a2 + 7a + 3




a2-5a + 2ab – 10b



Q. 3: Express each of the following as a fraction in its simplest form:



3m4 + 5m8 – m2



2p3 -3p +p2

Answer:

x=180-90-54, y=180-x,z=90

Step-by-step explanation:

the sum of the degree of a triangle will equal 180. the sum of the degree of a line will equal 180.

U(1,1), Q(4,4), and A(6,2) form a right triangle with UQ and QA being the legs and UA being the hypotenuse.

To determine whether U(1,1), Q(4,4), and A(6,2) form a right triangle, we will follow the given guide:

Find the slopes of sides UQ, QA, and UA:

Determine which segments are perpendicular and how we know they are perpendicular:

To determine if two lines are perpendicular, we need to check if their slopes are negative reciprocals of each other.

Find the lengths of each side:

Use the Pythagorean Theorem to show that it is a right triangle:

Since we have determined that UQ and QA are perpendicular, we can use the Pythagorean Theorem to show that it is a right triangle.

Since (Length of UA)² + (Length of QA)² = (Length of UQ)², we know that the triangle is a right triangle.

U(1,1), Q(4,4), and A(6,2) form a right triangle with UQ and QA being the legs and UA being the hypotenuse.

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The total number of students who were surveyed is 575

The graph likely displays a horizontal axis and a vertical axis.

In this graph, the horizontal axis may indicate two categories, such as "juniors" and "seniors," and the vertical axis shows the number of students in each category who responded "yes," "no," or "not sure."

To find the total number of students surveyed, we need to look for the bar that represents the entire group of juniors and seniors. Typically, this bar will be labeled as "total," "all," or something similar. Once we locate this bar, we can add up the number of students represented by the bar. The value will be displayed on the vertical axis, and it should correspond to the sum of the bars for juniors and seniors separately.

=> 100 + 200 + 125 + 150 = 575

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The percentage of body weight the dog lost is 9.2%. Mary would weigh 52.664 kg after losing the same percentage of body weight as her dog.

A) To find the percentage of body weight the dog lost, first, calculate the actual weight loss: 25 kg - 2.3 kg = 22.7 kg. Then, divide the weight loss (2.3 kg) by the original weight (25 kg) and multiply by 100 to get the percentage: (2.3 kg / 25 kg) * 100 = 9.2%.

B) If Mary lost the same percentage of her body weight as the dog did, she would lose 9.2% of her weight. To calculate this, multiply her original weight (58 kg) by the percentage (9.2%): 58 kg * 0.092 = 5.336 kg. Now, subtract this weight loss from her original weight to find her new weight: 58 kg - 5.336 kg = 52.664 kg. So, Mary would weigh 52.664 kg after losing the same percentage of body weight as her dog.

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If the surface area of cylinder A is 700cm², the surface area of cylinder B is 1612.8cm².

Since the two cylinders are similar, their dimensions are proportional to each other. Let the height and radius of cylinder A be h and r, respectively, and let the height and radius of cylinder B be kh and kr, respectively, where k is the scale factor.

Since cylinder B uses 72.8% more metal than cylinder A, we have:

Volume of cylinder B = 1.728 times the volume of cylinder A

Using the formula for the volume of a cylinder, we have:

π(kr)²(kh) = 1.728πr²h

Simplifying the equation, we get:

k³ = 1.728

k = 1.2

So the scale factor is 1.2. Therefore, the height and radius of cylinder B are 1.2 times those of cylinder A. Hence, we have:

Surface area of cylinder B = 2π(1.2r)² + 2π(1.2r)(1.2h)

= 2.304(2πrh)

= 2.304(700cm²)

= 1612.8cm²

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Based on the given information, we can set up an equation using the fact that the measures of angles m1 and m2 are equal in circle P.

m1 = m2

Substituting the given values:

9x + 5 = 4x + 35

Solving for x:

9x - 4x = 35 - 5

5x = 30

x = 6

Now that we have found the value of x, we can substitute it back into the expressions for m1 and m2 to find their measures:

m1 = 9x + 5 = 9(6) + 5 = 59

m2 = 4x + 35 = 4(6) + 35 = 59

Therefore, both angles have a measure of 59 degrees.

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Answer:

Step-by-step explanation:

OD. M = log(I/10,000)

How can the magnitude of an earthquake be calculated when its intensity is 10,000 times that of the reference earthquake?

The correct equation that calculates the magnitude, M, of an earthquake with an intensity 10,000 times that of the reference earthquake is option B: M = log(10,000/Io).

In the given equation M = log(I/IO), I represents the intensity of the earthquake being measured, and IO represents the intensity of the reference earthquake. Since the intensity of the earthquake in question is 10,000 times that of the reference earthquake, we substitute I with 10,000 times IO.

Therefore, the equation becomes M = log(10,000/Io), which is option B. This equation allows us to calculate the magnitude of the earthquake based on the relative intensity compared to the reference earthquake.

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Using the Pythagorean theorem, the value of x, rounded to the nearest tenth, is 11.5 units.

In the given right triangle ABC with right angle B, you are given the lengths of sides BC (8 units) and AC (14 units). You are asked to find the length of side AB (x units). To do this, you can use the Pythagorean theorem, which states that the square of the hypotenuse (AC) is equal to the sum of the squares of the other two sides (AB and BC).

So, the equation for this triangle is:

AC² = AB² + BC²

Plug in the given values:

14² = x² + 8²

196 = x² + 64

Subtract 64 from both sides:

132 = x²

Now, find the square root of 132:

x ≈ 11.5

So, the value of x, rounded to the nearest tenth, is 11.5 units.

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you will have to walk approximately 17.2 miles back home.

A right triangle is a triangle in which one of the angles measures 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.  the length of the hypotenuse. Right triangles have many important applications in mathematics, science, and engineering, particularly in trigonometry, which is the study of the relationships between the sides and angles of triangles.

You have formed a right triangle with legs of length 10 and 14. To find the length of the hypotenuse, which is the distance back to your starting point, we can use the Pythagorean theorem:

[tex]c^2 = a^2 + b^2[/tex]

[tex]c^2 = 10^2 + 14^2[/tex]

[tex]c^2 = 100 + 196[/tex]

[tex]c^2 = 296[/tex]

c = sqrt(296)

c ≈ 17.2

Therefore, you will have to walk approximately 17.2 miles back home.

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1. Write out the given equation: 8/10 + unknown fraction = 97/100

2. Subtract 8/10 from both sides of the equation to isolate the unknown fraction:

  8/10 + unknown fraction - 8/10 = 97/100 - 8/10

  Simplifying the left side: unknown fraction = 97/100 - 8/10

3. Convert both fractions to have a common denominator of 100:

  97/100 - 8/10 = 97/100 - 80/100

  Simplifying the right side: unknown fraction = 17/100

4. Therefore, the unknown fraction is 17/100.

So, the correct answer is "seventeen hundredths".

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The area of the circles are 1384.74 cm², 20601.54 cm² and 1589.625 cm²

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

Radii = 21 cm, 81 ft, 45/2 cm

The area of a circle is calculated as

Area = πr²

substitute the known values in the above equation, so, we have the following representation

Area = 3.14 * 21² = 1384.74

Area = 3.14 * 81² = 20601.54

Area = 3.14 * (45/2)² = 1589.625

Hence, the area of the circles are 1384.74 cm², 20601.54 cm² and 1589.625 cm²

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The height of the trapezoidal tabletop is approximately 59.5 centimeters.

To calculate the height of a trapezoid with the given measurements, we need to use the formula for the area of a trapezoid: A = (b1 + b2)h / 2, where A is the area, b1 and b2 are the lengths of the two bases, and h is the height.

In this problem, the area (A) is 7,021 square centimeters, the longer base (b1) is 131 centimeters, and the shorter base (b2) is 105 centimeters. Our task is to find the height (h).

First, let's plug the given values into the formula:

7,021 = (131 + 105)h / 2

Now, simplify the equation:

7,021 = 236h / 2

To solve for h, multiply both sides of the equation by 2:

14,042 = 236h

Finally, divide both sides by 236:

h ≈ 59.5

Thus, the height is approximately 59.5 centimeters.

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The cost of a plumber's one-hour visit is $51, determined by the linear equation C = 51d, where C is the cost and d is the duration of the visit in hours.

If a plumber's visit lasts for a certain duration, the cost can be determined using the equation

                           C = 51d

where C is the cost and d is the duration of the visit in hours.

In this case, the duration of the plumber's visit is given as 1 hour. Substituting d = 1 in the equation, we get

                             C = 51(1) = $51

as the cost of the plumber's visit.

Therefore, if the plumber's visit lasts for one hour, it would cost $51 according to the given equation.

This cost may vary if the duration of the visit changes, as it is directly proportional to the duration of the visit.

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2x - 3 would be 92 - 3 = 89 inches, which is the length of the table

The table needs to be 2 times longer than it is wide, so its length is 2 times its width, or 2x.

The table also needs to be 3 inches shorter than it is wide, so its width is x + 3 inches.

The area of the table is 4,608 square inches, so we can set up an equation:

2x(x + 3) = 4,608

Simplifying this equation:

2x²+ 6x = 4,608

Dividing both sides by 2:

x²+ 3x - 2,304 = 0

We can solve for x using the quadratic formula:

x = (-b ±  √(b² - 4ac)) / 2a

In this case, a = 1, b = 3, and c = -2,304. Substituting these values:

x = (-3 ±  √(3² - 4(1)(-2,304))) / 2(1)

Simplifying:

x = (-3 ±  √(9 + 9,216)) / 2

x = (-3 ±  √(9,225)) / 2

x = (-3 ± 95) / 2

x = 46 or x = -49

Since the width of the table cannot be negative, we can ignore the negative solution. Therefore, x needs to be 46 inches to fit the given requirements.

The length of the table is 2x, or 2(46) = 92 inches, and the width is x + 3, or 46 + 3 = 49 inches. The area is 92 * 49 = 4,508 square inches, which matches the given area requirement.

So, 2x - 3 would be 92 - 3 = 89 inches, which is the length of the table.

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We have shown that xn is bounded and monotone increasing, and its limit is √2. First, we will show that xn is bounded and monotone increasing by induction:

Base Case: For n = 1, we have x1 > 1, which is true.

Inductive Hypothesis: Assume that xn > 1 for some n = k and show that xn+1 > xn for n = k.

Inductive Step:

We have xn+1 := 2−1/xn

Since xn > 1, we have 1/xn < 1

Therefore, 2−1/xn > 2−1/1 = 1/2

So, xn+1 > 1/2

Since xn > 1, we have xn+1 = 2−1/xn < 2−1/1/ = 1

So, 1/2 < xn+1 < 1

Therefore, xn is bounded and monotone increasing.

Next, we will find the limit of xn as n → ∞:

Let L = lim xn as n → ∞

Then, taking the limit on both sides of xn+1 = 2−1/xn, we get:

L = 2−1/L

Multiplying both sides by L, we get:

L2 = 2−1

Solving for L, we get:

L = ±√2

Since xn > 1 for all n, we have L > 1. Therefore, L = √2.

Thus, we have shown that xn is bounded and monotone increasing, and its limit is √2.

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The backpack forms an angle of approximately 15 degrees with the ground.

What is Pythagorean theorem?

The Pythagorean theorem is a fundamental concept in mathematics that relates to the lengths of the sides of a right triangle.

To find the angle that Luis's backpack forms with the ground, we can use the inverse tangent function.

The height of the backpack when the handle is extended is 3.5 feet, and the distance from the ground to Luis's hand is 3 feet. So the opposite side of the triangle is 3.5 - 3 = 0.5 feet, and the adjacent side is the distance from Luis's hand to the backpack, which we can call x.

Using the tangent function, we have:

tan(theta) = opposite/adjacent

tan(theta) = 0.5/x

To solve for x, we can use the Pythagorean theorem:

x² + 3² = (3.5)²

x² = 3.5² - 3²

x² = 3.25

x = sqrt(3.25)

x ≈ 1.8 feet

Now we can substitute x into our tangent equation and solve for theta:

tan(theta) = 0.5/1.8

theta = arctan(0.5/1.8)

theta ≈ 15 degrees

Therefore, the backpack forms an angle of approximately 15 degrees with the ground.

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Note that Dawson's annual premium will be $2,462.40.

Dawson's annual premium will be $2,462.40.

This can be derived by going across from "Male 40-44" over to "20-year coverage" which is $13.68. Since $13.68 is per $1000 of coverage, you would multiply it by 180 to get $2,462.40.

An insurance premium is the amount of money paid by a person, firm, or enterprise to obtain an insurance coverage. The amount of the insurance premium is governed by a variety of factors and varies from one payee to the next.

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Full Question:

Although part of your question is missing, you might be referring to this full question:

The rate at which the cost of production is changing is 1000 hundred dollars per day or $100,000 per day.

To find the rate at which the cost of production is changing, we'll use the given cost function C = 50x + 1000, and the information that production is growing at a rate of 20 cases per day when x = 50 cases.

First, differentiate the cost function with respect to x to get the rate of change of the cost with respect to the number of cases produced (dC/dx):

dC/dx = 50

The derivative, 50, tells us that the cost increases by 50 hundred dollars for each additional case produced.

Now, we're given that dx/dt = 20 cases per day when x = 50 cases. To find dC/dt, the rate at which the cost of production is changing, multiply the rate of change of the cost with respect to the number of cases (dC/dx) by the rate of change of the number of cases with respect to time (dx/dt):

[tex]dC/dt = (dC/dx) × (dx/dt) = 50 × 20[/tex]

dC/dt = 1000

So, the rate at which the cost of production is changing is 1000 hundred dollars per day or $100,000 per day.

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The value of x is 5/3.

A fraction is a part of a whole. In arithmetic, the number is expressed as a quotient, in which the numerator is divided by the denominator.

We have equation in fraction are:

[tex]\frac{3x-2}{3x+1} = \frac{1}{2}[/tex]

To solve the value of x

In the above equation, Solve by cross multiplication:

2(3x - 2) = 3x + 1

Open the bracket and multiply by 2 :

6x - 4 = 3x +1

Combine the like terms:

6x - 3x = 1 + 4

Add and subtract the terms:

3x = 5

x = 5/3

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Proportionately, if 8 cm represents​ 10%, the length of a line segment that is​ 100% is 80 cm.

Proportion is the ratio of two quantities equated to each other.

Proportion also represents the portion or part of a whole.

Proportions can be represented using decimals, fractions, or percentages, like ratios.

The percentage of 8 cm length = 10%

The whole length = 100%

Proportionately, 100% = 80 cm (8 ÷ 10%) or (8 x 100 ÷ 10)

Thus, we can conclude that 100% of the line segment will be 80 cm if 8 cm is 10%.

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If 8 cm represents​ 10%, what is the length of a line segment that is​ 100%? Explain.

Answer: 8.75

Step-by-step explanation:

We know that 4 goes into 35 eight times.

        35 - (4 * 8) = 3

Next, we know that 3/4 is equal to 0.75 by dividing.

This leaves us with 8.75. Eight wholes and a part of 0.75.

Answer:

Based on the image, we can see that matrix A is a 3x2 matrix, and matrix B is a 2x3 matrix. In order to multiply matrices A and B, the number of columns in matrix A must match the number of rows in matrix B.

Since matrix A has 2 columns and matrix B has 2 rows, we can multiply them together, resulting in a 3x3 matrix. Therefore, the answer is B. The dimensions of AB are 3x3.

Ans. (c) 2X3

Dimension of matrix is given by row x column

The calculated t-value of 1.100 is less than the critical t-value of 2.718, we fail to reject the null hypothesis.

To test the hypothesis that regular physical exercise improves academic achievement, we will conduct a two-sample t-test for independent samples. The null hypothesis is that there is no difference in the mean academic achievement scores between the treatment group (physical exercise) and the control group (no physical exercise).

Let's calculate the mean and standard deviation for each group:

Treatment group (physical exercise):

mean = (4 + 2.67 + 3.65 + 2.11 + 3.21 + 3.6) / 6 = 3.3833

standard deviation = 0.7589

Control group (no physical exercise):

mean = (3.75 + 2.74 + 3.42 + 1.67 + 3 + 3.25 + 2.65) / 7 = 3.0071

standard deviation = 0.7037

We can now calculate the t-statistic:

t = (3.3833 - 3.0071) / sqrt((0.7589^2 / 6) + (0.7037^2 / 7)) = 1.100

The degrees of freedom for this test are 6 + 7 - 2 = 11 (assuming equal variances).

Using a t-table or a t-distribution calculator with 11 degrees of freedom and a significance level of 0.01, we find that the critical t-value is ±2.718.

Since the calculated t-value of 1.100 is less than the critical t-value of 2.718, we fail to reject the null hypothesis. We do not have enough evidence to conclude that regular physical exercise improves academic achievement.

The p-value for this test can be calculated as the probability of getting a t-value as extreme as 1.100, assuming the null hypothesis is true. Using a t-distribution calculator with 11 degrees of freedom, we find that the p-value is 0.294 (rounded to three decimal places).

Since the test result is not statistically significant (p > 0.01), we do not need to report an effect size using Cohen's d.

This test result could be reported in the literature as follows: "A two-sample t-test for independent samples was conducted to examine the effect of regular physical exercise on academic achievement, while controlling for academic aptitude. Six pairs of college students with similar GPAs were randomly assigned to either a treatment group that attended daily exercise classes or a control group. The mean academic achievement score for the treatment group was 3.3833 with a standard deviation of 0.7589, while the mean academic achievement score for the control group was 3.0071 with a standard deviation of 0.7037. The t-test result was not statistically significant (t(11) = 1.100, p = 0.294), indicating that there is not enough evidence to conclude that regular physical exercise improves academic achievement."

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Answer:

  C = (-1, -2)

  D = (2, 0)

Step-by-step explanation:

You want the coordinates of points C and D in parallelogram ABCD such that point E(0, 2) is the intersection of the diagonals. Given points are A(1, 6) and B(-2, 4).

The diagonals of a parallelogram bisect each other. This means the point of intersection of the diagonals is the midpoint of each:

  E = (A +C)/2 . . . . . . . . . . . . . . . E is the midpoint of AC

  C = 2E -A = 2(0, 2) -(1, 6)

  C = (-1, -2)

and

  D = 2E -B = 2(0, 2) -(-2, 4) . . . . . using the same pattern

  D = (2, 0)

The number of different menu combinations Mr. Carlos' family can choose is 60.

To find the total menu combinations, you need to use the multiplication principle. Since there are 3 choices of appetizers, 5 choices of entrees, and 4 choices of cake, you simply multiply these numbers together. Here's the step-by-step explanation:

1. Multiply the number of appetizer choices (3) by the number of entree choices (5): 3 x 5 = 15
2. Multiply the result (15) by the number of cake choices (4): 15 x 4 = 60

So, there are 60 different menu combinations possible for Mr. Carlos' family to choose for their reception.

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Amount that will be in the account after 2 years with daily compounding is $28,484.03.

Amount that will be in the account after 2 years with continuous compounding is $28,498.84.

The appropriate compound interest formula is:

[tex]A = P(1 + r/n)^{nt[/tex]

A is the amount.

P is the principal.

r is the annual interest rate.

n is the number of times the interest is compounded all year.

t is the number of years.

(a) For daily compounding (n = 365), we have:

A = 26000(1 + 0.0365/365)³⁶⁵*²
A = 26000(1 + 0.0001)⁷³⁰
A = 26000(1.0001)⁷³⁰
A = 28,484.03

Therefore, the amount that will be in the account after 2 years with daily compounding is $28,484.03.

(b) For continuous compounding, we have:

A = P[tex]e^{rt[/tex]

e is the mathematical constant almost equal to 2.71828.

A = 26000[tex]e^{0.0365*2[/tex]
A = 26000[tex]e^{0.073[/tex]
A = 28,498.84

Therefore, the amount that will be in the account after 2 years with continuous compounding is $28,498.84.

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The cost of a small cup of coffee is $2.97, and the cost of a piece of banana bread is $5.13.

Let x represent the cost of a small coffee and y represent the cost of a piece of banana bread. We know:

Cost of muffin: $3.51

y = x + $2.16

5(x + $3.51) + 2(x + y) = $48.60

Substitute y with x + $2.16:

5(x + $3.51) + 2(x + (x + $2.16)) = $48.60

Solve for x:

9x + $21.87 = $48.60

9x = $26.73

x = $2.97

Find y:

y = x + $2.16

y = $2.97 + $2.16

y = $5.13

A slice of banana bread costs $5.13, while a small coffee costs $2.97.

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p(x) = x + 5, and k = 2. The expression x^2 + 7x + 12 / (x + 2) can be written in the form p(x) + k / (x + 2) as:

x + 5 + 2 / (x + 2)

To express the given expression x^2 + 7x + 12 / (x + 2) in the form p(x) + k / (x + 2), we will perform polynomial division.

1. Divide the numerator (x^2 + 7x + 12) by the denominator (x + 2):

(x^2 + 7x + 12) ÷ (x + 2)

2. Perform long division:

       x + 5
       ________________
x + 2 | x^2 + 7x + 12
       - (x^2 + 2x)
       ________________
             5x + 12
             - (5x + 10)
       ________________
                 2

3. Write the result:

p(x) + k / (x + 2) = x + 5 + 2 / (x + 2)

So, p(x) = x + 5, and k = 2. The expression x^2 + 7x + 12 / (x + 2) can be written in the form p(x) + k / (x + 2) as:

x + 5 + 2 / (x + 2)

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The function represents exponential decay with a percentage rate of decrease of 9.8%.

The given exponential function y = 660(0.902) represents decay because the base of the exponent is less than one.

This means that the output value of the function will decrease as the input value increases.

To determine the percentage rate of decrease, we need to find the value of the base of the exponent subtracted from one and then multiply it by 100.

The base of the exponent is 0.902, so we subtract it from one to get 0.098.

Multiplying by 100 gives us a percentage rate of decrease of 9.8%.

This means that for every unit increase in the input value, the output value of the function will decrease by approximately 9.8%.

For example, if the input value increases from 1 to 2, the output value will decrease by 9.8%, and if the input value increases from 2 to 3, the output value will again decrease by 9.8%.

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