Of the following options, what could be a possible first step in solving the
equation -7x- 5 = x + 3? (6 points)
Adding 7x to both sides of the equation
O Subtracting 5 from both sides of the equation
Adding x to both sides of the equation
O Combining like terms, -7x + x = - 6x

The step which include the mistake is step 5.

The correct answer choice is option D.

[tex] \frac{1 + {3}^{2} }{5} + | - 10| \div 2[/tex]

Step 1:

[tex] = \frac{1 + {3}^{2} }{5} + 10 \div 2[/tex]

Step 2:

[tex] = \frac{1 + 9 }{5} + 10 \div 2[/tex]

Step 3:

[tex] = \frac{10}{5} + 10 \div 2[/tex]

Step 4:

[tex] = 2 + 10 \div 2[/tex]

Step 5:

[tex] = 12 \div 2[/tex]

Step 6:

[tex] = 6[/tex]

The step which include the mistake is step 5; because it didn't follow the rule of PEMDAS

P = parenthesis

E = exponents

M = Multiplication

D = Division

A = addition

S = subtraction

Therefore,

It should be;

[tex] = 2 + 5[/tex]

[tex] = 7[/tex]

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To find the probability of a contestant liking cats and dogs but not rabbits, we can use the formula for calculating the probability of independent events. That is, P(A and B and not C) = P(A) * P(B) * P(not C).

So in this case, P(cats and dogs and not rabbits) = 0.1 * 0.9 * 0.4 = 0.036. Therefore, the probability of a contestant liking cats and dogs but not rabbits is 0.036 or 3.6%.

As for the most likely outcome of this contestant's preferences, we can see that 90% of the contestants like cats, so it's very likely that this contestant likes cats. However, only 10% of the contestants like dogs, so it's less likely that this contestant likes dogs.

And 60% of the contestants like rabbits, so it's even more likely that this contestant does not like rabbits. Therefore, the most likely outcome is that this contestant likes cats but does not like dogs or rabbits.

In conclusion, given the probabilities provided, we can calculate the probability of a contestant liking cats and dogs but not rabbits, and we can also determine the most likely outcome of this contestant's preferences. The independence of the events allows us to use simple probability calculations to make these determinations.

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The students will need $74 on the third day.

Let x be the cost of one item at Store A and y be the cost of one item at Store B. Then the system of equations to model the situation is:

4x + 7y = 30
3x + 5y = 22

To find the cost on Day 3, we need to solve for x and y, and then use those values to calculate:

10x + 17y = ?

Part B: Solve the system of equations to find x and y
To solve the system of equations, we can use elimination or substitution. Here, we'll use substitution.

From the first equation, we can solve for x:

4x + 7y = 30
4x = 30 - 7y
x = (30 - 7y)/4

Substitute this expression for x into the second equation:

3x + 5y = 22
3((30 - 7y)/4) + 5y = 22
(90 - 21y)/4 + 5y = 22
90 - 21y + 20y = 88
-y = -2
y = 2

Now that we know y = 2, we can substitute this value back into either equation to find x:

4x + 7y = 30
4x + 7(2) = 30
4x + 14 = 30
4x = 16
x = 4

So x = 4 and y = 2.

Part C: Calculate the amount of money needed on Day 3
Finally, we can use these values to calculate the amount of money needed on Day 3:

10x + 17y = 10(4) + 17(2) = 40 + 34 = 74

Therefore, the students will need $74 on the third day.

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Answer:

5.63 degrees  to the nearest hundredth.

Step-by-step explanation:

Length of the plywood

= area / width

= 10.2 / 2

= 5.1 m

Sin x = 0.8 / 5.1    where x is the agle of elevation

sin x = 0.09804

x = 5.626 degrees

The duration of days after which the individual will run out of iced tea is 4 days, under the condition that  a pitcher can hold 13 cups of iced tea. The individual drinks 1. 75 cups of the tea every morning and 1. 5 cups of the tea each evening. 

So to solve this problem we have to relie on the basic principles of division
Total amount of tea consumed per day = 1.75 cups (morning) + 1.5 cups (evening)
= 3.25 cups
Total amount of tea in the pitcher = 13 cups
Number of days before running out of iced tea = 13 cups / 3.25 cups per day
= 4 days
Then, the duration of days until the iced tea runs out is 4 days.
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We may use the identity (a + b)³ = a³ + 3a²b + 3ab² + b³ to get 970299, 1061208, and 992016008 for (i), (ii), and (iii), respectively.

Using the identity (a + b)³ = a³ + 3a²b + 3ab² + b³ allows us to expand and simplify the expressions by distributing and collecting like terms. Any integer's cubes can be calculated using this method.

(i) Using the identity, we can write 99 as 100 - 1 and get:

= (99)³

= (100 - 1)³

= 100³ - 3(100²)(1) + 3(100)(1²) - 1³

= 970299

(ii) We can denote 102 as 100 + 2 and use the identity to obtain:

= (102)³

= (100 + 2)³

= 100³ + 3(100²)(2) + 3(100)(2²) + 2³

= 1061208

(iii) Using the identity, we may write 998 as 1000 - 2 and get:

= (998)³

= (1000 - 2)³

= 1000³ - 3(1000²)(2) + 3(1000)(2²) - 2³

= 992016008

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A real-world situation that might involve three linear equations in three variables is trying to determine the ages of three siblings.

Let's call them A, B, and C. We know that the sum of all three ages is a certain value, let's say it's 60. We also know the sum of the first two ages is either twice the third age or the third age squared. For example, if the sum of the first two ages is twice the third age, we could write it as A + B = 2C.

Alternatively, if the sum of the first two ages is the third age squared, we could write it as A + B = C^2. Similarly, we know the sum of the first and third ages is either twice the second age or the square root of the second age. So, we could write it as A + C = 2B or A + C = sqrt(B).

We now have three linear equations in three variables that we can use to solve for the ages of the three siblings. By solving the system of equations, we can find out how old each sibling is.

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The area of the composite figure is approximately 2447.43 square millimeters to the nearest hundredth.

To find the area of the composite figure, we need to divide it into simpler shapes and then find their areas separately. The composite figure is made up of a rectangle and two semicircles.

First, let's find the area of the rectangle. The length of the rectangle is 55 mm and the width is 32.5 mm, so the area of the rectangle is:

[tex]$$A_{rect} = length \times width = 55 \text{ mm} \times 32.5 \text{ mm} = 1787.5 \text{ mm}^2$$[/tex]

Next, let's find the area of each semicircle. The diameter of each semicircle is equal to the width of the rectangle, which is 32.5 mm. Therefore, the radius of each semicircle is:

[tex]$$r =[/tex] [tex]\frac{32.5 \text{ mm}}{2} = 16.25 \text{ mm}$$[/tex]

The formula for the area of a semicircle is:

[tex]$$A_{semicircle} = \frac{1}{2} \pi r^2$$[/tex]

So, the area of each semicircle is:

[tex]$$A_{semicircle} = \frac{1}{2} \pi (16.25 \text{ mm})^2 \approx 329.97 \text{ mm}^2$$[/tex]

To find the total area of the composite figure, we add the area of the rectangle to the area of the two semicircles.

[tex]$$A_{total} = A_{rect} + 2 \times A_{semicircle} \approx 2447.43 \text{ mm}^2$$[/tex]

Therefore, the area of the composite figure is approximately 2447.43 square millimeters to the nearest hundredth.

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The statements that are always true for geometric shapes are:

1) Always

2) Always

3) Sometimes

4) Never

1) A square is a type of rectangle in which all four sides are equal. Therefore, all of the properties that apply to rectangles (such as having four right angles and opposite sides that are parallel) also apply to squares, making the statement "A square is a rectangle" always true.

2) The diagonals of a rhombus are always perpendicular to each other. This is because a rhombus has opposite sides that are parallel, and the diagonals bisect each other at a right angle.

3) The diagonals of a rectangle are sometimes equal. This is true only if the rectangle is a square (where all four sides are equal) or if the rectangle is a "golden rectangle" (where the ratio of the longer side to the shorter side is equal to the golden ratio).

4) The diagonals of a trapezoid are never equal unless the trapezoid happens to be an isosceles trapezoid (where the legs are equal in length). In general, the diagonals of a trapezoid will have different lengths, and there is no special relationship between them.

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Answer:  D.  7/15 or

Step-by-step explanation:

You have 15 possible outcomes

Probability= possibilities/outcomes

Possible numbers that are multiples of 3 are: 3, 6, 9, 12, 15.  There are 5 possibilities.

P(multiple of 3) = 5/15

Possible numbers that are multiples of 7 are: 7, 14, .  There are  2 possibilities.

P(7) = 2/15

Because of the or you add th e probabilities

P(multiple of 3 or multiple of 7) = 5/15 +2/15 =7/15

D

By using completing the square, the equation can be rewritten as (x + (-11/4))² = 9/16.

In Mathematics and Geometry, the standard form of a quadratic equation is represented by the following equation;

ax² + bx + c = 0

Next, we would solve the given quadratic equation by using the completing the square method;

2x² - 11x + 14 = 0

By dividing all through by 2, we have:

x² - 11x/2 + 7 = 0

x² - 11x/2 = -7

In order to complete the square, we would have to add (half the coefficient of the x-term)² to both sides of the quadratic equation as follows:

x² - 11x/2 + (-11/4)² = -7 + (-11/4)²

x² - 11x/2 - 121/16 = 9/16

(x - 11/4)² = 9/16.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The interest gained by Lesli is if she lends $5000 for 3 years at 5% simple interest annually is $750, 10% simple interest annually is $1500, and on 5% compound interest annually is $790

The simple interest is calculated by

I = P * r * t

where P is the principal

r is the rate of interest

t is the time

I is the simple interest

The compound interest is calculated by:

I = P[tex](1 +r)^t[/tex] - P

where I is the compound interest

P is the principal

r is the rate of interest

t is the time

According to the question,

P = $5000

t = 3 years

1. r = 5% simple interest

I = 5000 * 3 * 0.05

= $750

2. r = 10% simple interest

I = 5000 * 3 * 0.10

= $1500

3. r = 5% compounded annually

I = 5000 [tex](1+0.05)^3[/tex] - 5000

= 5000 * 1.158 - 5000

= $790

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The question is in Spanish, and the question in English is:

How much interest will Lesli earn if she lends 15,000 to be paid in 3 years? at: 5% annual simple. 10% simple annual. 5% compounded annually

The finance charge on a credit card account with a balance of $660.30 and an APR of 6.2% is $3.41.

To calculate the finance charge on a credit card account with a balance of $660.30 and an APR of 6.2%. Here's a step-by-step explanation:

1. Convert the APR (Annual Percentage Rate) to a decimal by dividing it by 100: 6.2 / 100 = 0.062

2. Divide the APR decimal by 12 to find the monthly interest rate: 0.062 / 12 = 0.005167

3. Multiply the credit card balance by the monthly interest rate: $660.30 * 0.005167 = $3.41

The finance charge on a credit card account with a balance of $660.30 and an Annual Percentage Rate (APR) of 6.2% is determined to be $3.41. This finance charge represents the cost of borrowing on the credit card and is calculated based on the outstanding balance and the interest rate.

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The null hypothesis (H0) for this situation would be that there is no difference in the proportion of male and female students with an overall GPA of 3.5 or higher.

The alternative hypothesis (Ha) would be that there is a difference in the proportion of male and female students with an overall GPA of 3.5 or higher.

Formally, we can write the hypotheses as:

H0: p_male = p_female (where p_male represents the proportion of male students with an overall GPA of 3.5 or higher, and p_female represents the proportion of female students with an overall GPA of 3.5 or higher)

Ha: p_male ≠ p_female

The president of the university can test these hypotheses using a hypothesis test for the difference in proportions. She can calculate the test statistic using the sample proportions and sample sizes for male and female students, and then compare it to the appropriate critical value or p-value based on the desired level of significance.

If the test results provide strong evidence against the null hypothesis, she can reject it and conclude that there is a statistically significant difference in the proportion of male and female students with an overall GPA of 3.5 or higher. If the test results do not provide enough evidence to reject the null hypothesis, she can fail to reject it and conclude that there is not enough evidence to suggest a difference in grades between males and females.

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1/20 of the customers are dairy intolerant.

We know that 1/5 of the customers are vegetarian, and 3/4 eat meat. Let's first find the total fraction of vegetarian and meat-eating customers:

1/5 (vegetarian) + 3/4 (meat)

To add these fractions, we need a common denominator. The least common denominator (LCD) for 5 and 4 is 20. So, we'll convert both fractions to have the same denominator:

(1/5)*(4/4) = 4/20 (vegetarian)
(3/4)*(5/5) = 15/20 (meat)

Now, let's add the fractions:

4/20 (vegetarian) + 15/20 (meat) = 19/20 (vegetarian and meat)

Now we know that 19/20 of the customers are either vegetarian or meat-eaters. The remainder must be dairy intolerant. To find this fraction, subtract the combined fraction from 1:

1 - 19/20 = 1/20

So, 1/20 of the customers are dairy intolerant.

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Answer:

To find the weighted mean, we need to multiply each delivery value by its corresponding frequency, add the products, and divide by the total frequency.

(3 x 7) + (6 x 6) + (9 x 1) + (12 x 0) = 21 + 36 + 9 + 0 = 66

Total frequency = 7 + 6 + 1 + 0 = 14

Weighted mean = 66 / 14 = 4.7 (rounded to the nearest tenth)

Therefore, the weighted mean is 4.7

The value of the x is 8.33 under the given condition that  RZ is given as 2x + 5 and TW is given as 5x - 20.

From the given question and illustrative diagram we can clearly see that
RZ = 2x + 5
TW = 5x - 20
Now, we have to find the value of 'x' if RZ = 2x + 5 and TW = 5x - 20.
Then, from the given rectangle figure, we can say  that RZ is equal to TW.
Hence equating both the equation we can  evaluate that the value of x  and the equation can be expressed in the forms of
RZ = TW
2x + 5 = 5x - 20
20 + 5 = 5x - 2x
25 = 3x
x = 25/3
x = 8.33
Then, the value of the x is 8.33.
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The value of the tangent of J, tan J = 6.2/4

To determine the value, we need to find the opposite side of the angle J.

Using the Pythagorean theorem, we have that;

(√55)² = 4² + j²

Find the square of the values, we get;

55 = 16+ j²

collect the like terms, we have;

j² = 55 - 16

subtract the values

j² = 39

Find the square root of both sides

j = 6. 2

Then, using the tangent identity, we have;

tan J = opposite/adjacent

Opposite = 6. 2

Adjacent = 4

Substitute the values

tan J = 6.2/4

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The error in the calculation is that n/360 should be equal to the central angle of the sector in degrees divided by 360. However, the given value of 115/225 is not the correct central angle. To find the correct central angle, we need to use the formula for the area of a sector:

Area of sector XZY = (central angle/360) x πr^2

We know that the area of circle ⊙Z is 255 square feet, so we can find the radius:

πr^2 = 255
r^2 = 81.11
r ≈ 9 feet

Now we can solve for the central angle:

Area of sector XZY = (central angle/360) x π(9)^2
Area of sector XZY = (central angle/360) x 81π
Area of sector XZY = (central angle/360) x 254.47

Since the area of sector XZY is not given, we cannot use the given equation n/360 = 115/225 to find the central angle. Instead, we need to use the formula above and solve for the central angle. Let A be the area of sector XZY:

A = (n/360) x 254.47
n/360 = A/254.47
n = 360A/254.47

Now we can substitute the given area of circle ⊙Z and solve for the area of sector XZY:

255 = (n/360) x πr^2
255 = (n/360) x π(81)
255 = (n/360) x 254.47
n = (360 x 255)/254.47
n ≈ 360.15

Note that n should be rounded to the nearest integer since it represents the central angle in degrees. Therefore, the central angle is approximately 360 degrees. Now we can use this value to find the area of sector XZY:

Area of sector XZY = (360/360) x π(9)^2
Area of sector XZY = 81π
Area of sector XZY ≈ 254.47 ft^2

Therefore, the area of sector XZY should be approximately 254.47 square feet.

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The correct equation that gives the population, a(x), of the town x years since 2005 is:

a(x) = 12,500 * (1 + 0.054)ˣ

The given problem states that the population of the town has been increasing by about 5.4% per year since 2005. To formulate the equation for the population, we need to use the initial population of 12,500 in 2005 and apply the growth rate of 5.4% per year.

The general formula for exponential growth is:

a(x) = a(0) * (1 + r)ˣ

Where:

a(x) is the population at a given time x years since the initial time,

a(0) is the initial population (12,500 in this case),

r is the growth rate (5.4% or 0.054 as a decimal),

x is the number of years since the initial time (2005 in this case).

Plugging in the values, we get:

a(x) = 12,500 * (1 + 0.054)ˣ

This equation calculates the population of the town x years since 2005.

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Answer: is approximately between 113.39 and 116.61
To calculate the 95% confidence interval for the students' average IQ score, we'll use the given information: sample size (n=150), sample mean (X=115), and sample standard deviation (s=10). We'll use the t-distribution since the population standard deviation is unknown.

First, we need to find the t-value for a 95% confidence interval with n-1 (149) degrees of freedom. Using a t-table or calculator, we find the t-value to be approximately 1.976.

Next, we'll calculate the standard error (SE) using the formula: SE = s/√n. In this case, SE = 10/√150 ≈ 0.816.

Now, we can find the margin of error (ME) using the formula: ME = t-value × SE. For this problem, ME = 1.976 × 0.816 ≈ 1.61.

Finally, to calculate the 95% confidence interval, we'll use the formula: X ± ME. Thus, the 95% confidence interval is 115 ± 1.61, which is approximately (113.39, 116.61).

So, the 95% confidence interval for the students' average IQ score is approximately between 113.39 and 116.61.

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Answer:

see below

Step-by-step explanation:

Here are the steps that Frank took:

1/5t+2=17

1/5t+2-2=17

1/5t=17

t=85

Frank is incorrect.  He is incorrect because in step 2, he forgot to subtract both sides by 2, and only did this to the left side of the equal sign.  He has to subtract 2 from both sides of the equal side to have the equation remain balanced.  Frank should've gotten t=75.

Hope this helps! :)

Answer:

$450

Step-by-step explanation:

Let's use the variables C, I, and F to represent the number of chocolate fudge bars, ice cream sandwiches, and frozen fruit bars, respectively, that you will sell.

The cost of each chocolate fudge bar is $0.75, the cost of each ice cream sandwich is $0.85, and the cost of each frozen fruit bar is $0.50. Therefore, the total cost of the frozen treats that you buy will be:

Total cost = 0.75C + 0.85I + 0.50F

We want to make sure that this total cost is no more than $450. Therefore, we can write the following inequality:

0.75C + 0.85I + 0.50F ≤ 450

This inequality represents the number of chocolate fudge bars, C, the number of ice-cream sandwiches, I, and the number of frozen fruit bars, F, that will cost you no more than $450.

The parametric equations to plot points. The sixth point is (0, 1).

The parametric equations given are:

x = sin t

y = 1 − cos t

To plot points for these equations, we can choose some values of t and substitute them in the equations to get the corresponding values of x and y. Here are some points we can plot:

When t = 0, x = sin 0 = 0 and y = 1 − cos 0 = 1.

So the first point is (0, 1).

When t = π/4, x = sin (π/4) = √2/2 and y = 1 − cos (π/4) = 1 − √2/2.

So the second point is (√2/2, 1 − √2/2).

When t = π/2, x = sin (π/2) = 1 and y = 1 − cos (π/2) = 0.

So the third point is (1, 0).

When t = π, x = sin π = 0 and y = 1 − cos π = 2.

So the fourth point is (0, 2).

When t = 3π/2, x = sin (3π/2) = −1 and y = 1 − cos (3π/2) = 0.

So the fifth point is (−1, 0).

When t = 2π, x = sin 2π = 0 and y = 1 − cos 2π = 1.

So the sixth point is (0, 1).

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The height of the ice block must be 10 inches, the length and width could be any combination of dimensions that multiply together to equal 48 square inches.

The overall number of cube units that the cube totally occupies is the definition of a cube's volume. Volume is simply the total amount of space an object takes up. The cube's volume can be calculated using the formula a3 where an is the cube's edge.

given,

To check if the height of the ice block must be 10 inches to have a volume of 480 cubic inches, we can use the formula for the volume of a rectangular solid:

V = l * w * h

where l represents the length, w represents the measurement of width, while h is the peak, and V is the volume.

Since the volume is given as 480 cubic inches, and the height is specified as 10 inches, we can write:

480 = l * w * 10

Dividing both sides by 10, we get:

48 = l * w

This means that the product of the length and width must be equal to 48 square inches in order for the block of ice to have a volume of 480 cubic inches with a height of 10 inches.

There are many possible dimensions that satisfy this condition. For example, the block of ice could have dimensions of 8 inches by 6 inches by 10 inches, or 12 inches by 4 inches by 10 inches, or 16 inches by 3 inches by 10 inches, and so on.

Therefore, while the height of the ice block must be 10 inches, the length and width could be any combination of dimensions that multiply together to equal 48 square inches.

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A) The null hypothesis (H0) is that there has been no change in the prevalence of HIV since the last census. The alternative hypothesis (Ha) is that there has been a change in the prevalence of HIV since the last census

B) At a 95% confidence level, the critical value for a two-tailed test is ±1.96.

A) The null hypothesis (H0) is that there has been no change in the prevalence of HIV since the last census, i.e., the current prevalence of HIV is still 7.5%. The alternative hypothesis (Ha) is that there has been a change in the prevalence of HIV since the last census, i.e., the current prevalence of HIV is different from 7.5%.

B) To test the hypothesis, we can use a z-test for proportions. The test statistic can be calculated as:

z = (p - p0) / [tex]\sqrt{(p0(1-p0)/n)}[/tex]

where p is the sample proportion of HIV positive cases, p0 is the hypothesized proportion under the null hypothesis, and n is the sample size.

In this case, p = 468 / 4706 = 0.099, p0 = 0.075, and n = 4706. Plugging these values into the formula, we get:

z = (0.099 - 0.075) / [tex]\sqrt{0.075(1-0.075)/4706}[/tex] = 8.80

At a 95% confidence level, the critical value for a two-tailed test is ±1.96. Since the calculated z-value (8.80) is much larger than the critical value, we can reject the null hypothesis and conclude that there is strong evidence to suggest that the prevalence of HIV has changed since the last census. In other words, the prevalence of HIV is different from 7.5%.

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The 95% confidence interval is: b) -1.38 to d) 3.44.

The value of the sample mean difference is 1.74, which falls:

b) within.

Here, we have to determine the 95% confidence interval for the difference of sample means and complete the statements, we need to use the sample mean difference provided and the confidence interval limits given as options.

We'll compare the sample mean difference to the interval to see if it falls within or outside the interval.

Given that the sample mean difference is 1.74, let's analyze the options:

Options for the confidence interval limits:

Lower limit options:

a) -1.26

b) -1.38

c) -3.48

d) -3.44

Upper limit options:

a) 1.26

b) 3.48

c) 1.38

d) 3.44

Since the sample mean difference is 1.74, we need to check if it falls within the interval formed by the lower and upper limits.

Looking at the options for the lower limit, the closest value to 1.74 is -1.38, and the closest value to the upper limit is 3.44.

So, the 95% confidence interval would be:

-1.38 to 3.44

Now, completing the statements:

The 95% confidence interval is: b) -1.38 to d) 3.44

The value of the sample mean difference is 1.74, which falls:

b) within

So, the completed statements are:

The 95% confidence interval is -1.38 to 3.44.

The value of the sample mean difference is 1.74, which falls within the 95% confidence interval.

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Answer:

40%

Step-by-step explanation:

[tex] \frac{48}{120} \times 100 = 40[/tex]

Answer:

40%

Step-by-step explanation:

To find out what percentage of 2 hours is 48 minutes, we need to first convert both values to the same unit of time, such as minutes.

2 hours is equal to 120 minutes (2 x 60).

So, the fraction of 2 hours that is represented by 48 minutes is:

48/120

Simplifying this fraction by dividing both the numerator and denominator by 12, we get:

4/10

Multiplying the numerator and denominator by 10 to convert this fraction into a percentage, we get:

40%

Therefore, 48 minutes is 40% of 2 hours.

Answer:

25

Step-by-step explanation:

all three angles of a triangle equal to 180º. 135+20=155

180-155=25

this is how u get ur answer

135° + 20° = 155°

Since the angles must have a sum of 180°, we subtract 155° from 180°, which would equal 25°.

To confirm: 135° + 20° + 25° = 180°

To find the volume of the solid above the triangle R and below the plane z = x, we can use a triple integral with cylindrical coordinates.

First, we can note that the triangle R lies in the x-y plane and is symmetric with respect to the line y = x. Therefore, we can consider the solid above the portion of R in the first quadrant and then multiply the result by 4 to get the total volume.

In cylindrical coordinates, we have:

z = r cos(theta)

x = r sin(theta)

The bounds for r and theta can be obtained by considering the equations of the lines that bound the portion of R in the first quadrant. These lines are:

y = (1/2) x

y = 4 - (1/2) x

Solving for x and y in terms of r and theta, we get:

x = r sin(theta)

y = r cos(theta)

Substituting these expressions into the equations of the lines and solving for r, we get:

r = 8 sin(theta) / (3 + 2 cos(theta))

The bounds for theta are 0 and pi/2, since we are considering the portion of R in the first quadrant.

The bounds for z are from z = 0 to z = x = r sin(theta).

Therefore, the triple integral for the volume is:

V = 4 * ∫[0, pi/2] ∫[0, 8 sin(theta) / (3 + 2 cos(theta))] ∫[0, r sin(theta)] 1 dz dr dtheta

This integral can be evaluated using standard techniques, such as trigonometric substitution. The result is:

V = 32/3

Therefore, q5 = 32/3.

Learn more about trigonometric substitution.

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