Prove that there exists infinitely many integers $n$ such that $n^2+1$ is squarefree. (a square-free integer is an integer which is divisible by no square number other than 1.)

Answer:

We should use 30 pints of the first drink and 120 pints of the second drink

Step-by-step explanation:

Let x represent the number of pints of the first drink used
Volume of fruit juice in x pints = 35% of x = 0.35x

Let y represent the number of pints of the second drink used
Volume of fruit juice in y pints = 85% of y= 0.85y

Since the final mixture volume is 150 pints we get
x + y = 150  ..... [1]

If the final mixture has to contain 75% pure fruit juice, the volume of fruit juice in the mixture = 0.75 x 150 = 112.5 pints

Therefore
0.35x + 0.85y = 112.5  ...[2]

We will solve this set of simultaneous equations by making the coefficients of one of the variables equal and then subtracting to eliminate that variable

Multiply equation [1] by 0.35 to get the x coefficients equal

0.35 x [1]
=>  0.35(x + y) = 0.35(150)

=> 0.35x + 035y = 52.5  ... [3]

Subtract [3] from [2]:

[3] - [2] =>

0.35x + 0.85y - (0.35x + 035y) = 112.5 - 52.5

0.35x + 0.85y - 0.35x - 0.35y = 60

0x + 0.5y = 60

0.5y = 60

y = 60/0.5

y = 120

Therefore x = 150 - 120 = 30

Therefore we should use 30 pints of the first drink and 120 pints of the second drink

Step-by-step explanation:

See image

The merchant needs to mix 10 pounds of red beans and 10 pounds of black beans to get a mix that costs $3.75 per pound.

We have,

Let's assume that the merchant needs to mix x pounds of red beans and (20 - x) pounds of black beans.

To find the pounds of each variety, we can use the following equation:

= 3.5x + 4(20 - x)

= 3.75(20)

Simplifying the equation.

3.5x + 80 - 4x = 75

-0.5x = -5

x = 10

Therefore,

The merchant needs to mix 10 pounds of red beans and 10 pounds of black beans to get a mix that costs $3.75 per pound.

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The complete question.

A merchant wants to mix red beans that cost $3.5 per pound with black beans that cost $4 per pound to get 20 pounds of a mix that costs $3.75 per pound. How many pounds of each variety should he mix?

The calculated value of x from the special triangle is 17.0

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

The right triangle

The right triangle is a special triangle that has a measure of 45 degrees

For a triangle that has a measure of 45 degrees, we have

Hypotenuse = Legs√2

In this case, we have

Leg = x

Hypotenuse = 24

So, we have

x√2 = 24

Divide by √2

x = 24/√2

Evaluate

x = 17.0

Hence, the calculated value of x is 17.0

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question 9.

Based on a cubic modeling equation, the  prediction about this data that is  most reliable is a $70k house would have 1,354 square ft.

Option A is correct.

Question 10.

The second coordinate in a scatterplot of the linearized data and  rounding the answer to the tenth place is 0.8

Option A is correct.

Question 11.

The approximate value when x=4 and rounding the answer to the nearest whole number is 1,168

Option C is correct.

The equation is given as

Log y= 0.75821x+0.03442

We substitute x = 4 into the given equation and solve for y:

log y = 0.75821(4) + 0.03442

log y = 3.07286 + 0.03442

log y = 3.10728

We take the antilogarithm of both sides,

y = 10^(3.10728)

y = 1,168.66

Rounding off, the closest answer is option  (c) 1,168.

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7. The exact values of x is 17.

8. The exact values of x is 12.

9. The exact values of x is 10.9.

10. The exact values of y is 15.

11. The exact values of y is 15.

12. The exact values of y is 6.

The exact values of x and y is calculated by applying Pythagoras theorem as follows;

7.

x² = 8² + 15²

x² = 289

x = √289

x = 17

8.

x² = 15² - 9²

x² = 144

x = √144

x = 12

9.

x² = 12² - 5²

x² = 119

x = √119

x = 10.9

The value of y is calculated as follows;

10.

y² = 12² + 9²

y² = 225

y = √(225)

y = 15

11.

y² = 17² - 8²

y² = 225

y = √(225)

y = 15

12.

y/10 = 6/10

y = 6

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this will be the square -

[tex]2 \times \sqrt[5]{4} [/tex]

or 2,63902

It will take Ms. Estrada about 25 hours to complete the mural.

Algebra is a branch of mathematics that deals with the study of mathematical symbols and their manipulation. It involves the use of letters, symbols, and equations to represent and solve mathematical problems.

In algebra, we use letters and symbols to represent unknown quantities and then use mathematical operations such as addition, subtraction, multiplication, division, and exponentiation to manipulate those quantities and solve equations. We can use algebra to model and solve real-world problems in various fields such as science, engineering, economics, and finance.

Some common topics in algebra include:

Solving equations and inequalities

Simplifying expressions

Factoring and expanding expressions

Graphing linear and quadratic functions

Using logarithms and exponents

Working with matrices and determinants

First, we need to convert the length of the mural from feet to inches to match the unit of the sections.

20 feet x 12 inches/foot = 240 inches

Then, we need to determine the number of sections in the mural:

240 inches / 8 inches/section = 30 sections

Now, we can find the total time it will take Ms. Estrada to complete the mural:

30 sections x 50 minutes/section = 1500 minutes

Finally, we can convert the time to hours:

1500 minutes / 60 minutes/hour = 25 hours

Therefore, it will take Ms. Estrada about 25 hours to complete the mural.

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Answer:

we have to first start converting the given volume to liters, since the answer is expected in hectoliters.

36 liters + 4,000 milliliters = 36 liters + 4 liters = 40 liters

1 hectoliter = 100 liters

Therefore, we have:

40 liters = 0.4 hectoliters

So, we have 0.4 hectoliters of fuel.

Answer:

0.4 hectoliters of fuel

Step-by-step explanation:

We Know

You have 36 liters of fuel and just added 4,000 milliliters more.

How many hectoliters of fuel do you have?

1000 milliliters = 1 liters

4000 milliliters = 4 liters

We Take

36 + 4 = 40 liters of fuel

We Convert 40 liters to hectoliters

1 liter = 0.01 hectoliters

40 liters = 40 x 0.01 = 0.4 hectoliters

So, you have 0.4 hectoliters of fuel.

Answer:

To calculate the monthly payments for each option, we can use the PMT function in Excel or a financial calculator.

For the new car option:

Loan amount = $31,000

Interest rate (annual) = 7.19%

Loan term = 3 years

Number of payments = 3 x 12 = 36

Using the PMT function in Excel, we get:

PMT(7.19%/12, 36, -$31,000) = $966.59

For the used car option:

Loan amount = $16,000

Interest rate (annual) = 5.89%

Loan term = 4 years

Number of payments = 4 x 12 = 48

Using the PMT function in Excel, we get:

PMT(5.89%/12, 48, -$16,000) = $385.36

The difference in monthly payments between the two options is:

$966.59 - $385.36 = $581.23

Therefore, the difference in monthly payments between financing the new car and financing the used car is $581.23.

Step-by-step explanation:

Answer:

vertical because they are right across from each with the same degrees

Using the quadratic equation, the solution is x = -8 and x = -17.

A second-order equation of the form ax2 + bx + c = 0 denotes a quadratic equation, where a, b, and c are real number coefficients and a 0.

The equation is x²+25x+136=0

where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.

For the given equation x² + 25x + 136 = 0, we can identify a = 1, b = 25, and c = 136.

Substituting these values in the quadratic formula, we get:

x = (-25 ± √(25² - 4(1)(136))) / 2(1)

x = (-25 ± √(625 - 544)) / 2

x = (-25 ± √(81)) / 2

imply them

x = (-25 + 9) / 2 or x = (-25 - 9) / 2

x = -8 or x = -17

Therefore, the solutions to the quadratic equation x^2 + 25x + 136 = 0 are x = -8 and x = -17.

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Answer:

3 divide 2/5 is 7.5 since 2/5 is 0.4

If points A, B, and C are not three nonadjacent vertices of the cube or three adjacent vertices of the cube, then the cross-section will be a polygon with more than four sides, and its shape will depend on the location of the points A, B, and C on the cube.

What is meant by cross section?

The form we obtain by cutting a straight through an object is called a cross-section.

The shape of the cross section of a cube that is cut into two pieces by a single slice that passes through point A, B, and C depends on the location of these points on the cube.

If points A, B, and C are three nonadjacent vertices of the cube, then the cross section will be a triangle. Specifically, it will be an isosceles triangle with base equal to the length of the edge of the cube and legs equal to the distances between point A and the other two points B and C.

If points A, B, and C are three adjacent vertices of the cube, then the cross section will be a square. Specifically, it will be a square with side length equal to the length of the edge of the cube.

If points A, B, and C are not three nonadjacent vertices of the cube or three adjacent vertices of the cube, then the cross section will be a polygon with more than four sides, and its shape will depend on the location of the points A, B, and C on the cube.

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If in sample of 210 students from district and finds 35 have submitted "box-tops" this year, then

(a) Yes, data provide enough evidence regarding contest which has increased participation in "box-tops" for "education-program",

(b) the p-value is 0.0187.

Part(a) : To find if contest has increased the participation in "box-tops" for "education-program", we conduct a "hypothesis-test".

We use a "significance-level" of 0.05, which means we will accept a 5% chance of making a "Type I Error",

Let Null Hypothesis ⇒ The proportion of students submitting "box-tops" is = 12%,

Let Alternative Hypothesis ⇒ The proportion of students submitting "box-tops" for education is increased after contest,

To test above hypothesis, we use "one-sample" proportion-test.

We will use the sample proportion (p') = 35/210 = 0.1667, as estimate of  population proportion (p).

So, "test-statistic" (z) is ⇒ z = (p' - p₀)/√(p₀ × (1-p₀)/n),

where n = sample size,

In "Null-Hypothesis", the "expected-value" of the test statistic is 0, and  "standard-deviation" of the "test-statistic" is √(p₀×(1-p₀)/n),

Substituting, p₀ = 0.12 and n = 210,

We get,

⇒ z = (0.1667 - 0.12)/√(0.12 x 0.88/210) ≈ 2.08,

Now, we know that "right-tailed" critical value at 0.05 level is = 1.645,

Since, z (2.08) > 1.645, the Null-Hypothesis is Rejected.

So, above data provides enough evidence regarding contest has increased the participation in "box-tops" for "education-program".

Part(b) : We need to find the "p-value" for the test-statistic that we found in part(a),

So, p-value = P(z>2.08),

⇒ 1 - P(z ≤ 2.08),

⇒ 1 - 0.9813,

⇒ 0.0187,

Therefore, the required "p-value" is 0.0187.

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Answer:

Both solutions are valid, and depending on the context of the problem, one or both solutions may be useful in solving for different parameters, such as the maximum height or the time of flight of the projectile.

Step-by-step explanation:

In a projectile motion formula, such as the one that describes the height of an object in motion as a function of time, there are two solutions because the object can reach the same height at two different times during its trajectory.

The negative solution represents the time it takes for the object to reach its maximum height and then start falling back down to the ground. The positive solution represents the time it takes for the object to reach the same height on its way up.

So, both solutions are valid, and depending on the context of the problem, one or both solutions may be useful in solving for different parameters, such as the maximum height or the time of flight of the projectile.

After 16 minutes of decay, approximately 182.3 grams of the element remains.

To calculate how much of the element remains after 16 minutes of decay, we can use the formula for exponential decay:

A = A₀ * [tex]e^{(-kt)[/tex]

where A is the amount of the element remaining after time t, A₀ is the initial amount of the element, k is the decay constant, and t is the elapsed time.

In this case, we have A₀ = 420 grams, k = 0.118 (since the element decays by 11.8% per minute), and t = 16 minutes. Plugging these values into the formula, we get:

A = 420 * [tex]e^{(-0.118*16)[/tex]

A = 182.3 grams

We round our answer to the nearest 10th of a gram, as requested in the problem.

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The correct answer is option B. That is the amount of plastic needed for making one plastic cylinder is 57 cm²

A cylinder is a three-dimensional geometric shape that has two congruent circular bases that lie parallel to each other and are connected by a curved surface. It can be thought of as a stack of circles or as a prism with a curved surface. Some common examples of cylinders include soda cans, water bottles, and pipes.

Given that the individual container of bubbles will be a plastic cylindrical tube with a diameter 2 cm and a height of 8 cm.

So radius = diameter/2 = 2/2 = 1 cm.

The surface area of the cylinder = 2πrh + 2πr²

SA = 2π × 1 ×8 + 2π × 1²

SA = 2π + 16π

SA = 18π

Substituting the value of π = [tex]\frac{22}{7}[/tex]

Hence, SA = 18π = 18 × [tex]\frac{22}{7}[/tex] ≈ 56.57

Rounding to the nearest integer, we get:

Answer: B. 57 cm^2.

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The height of the building , given the shadow that it casts would be 93 meters tall .

Utilizing proportions can be an effective method to tackle this issue. By defining ' h ' as the height of the building responsible for casting a 39.75 meter shadow, we may establish the ensuing proportion :

h / 39. 75 = 3. 2 / 1. 37

Solving for h gives us :

h / 39. 75 = 2. 336

h = 2. 336 x 39. 75

h = 92. 8 56

h = 93 meters tall

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Answer:

Step-by-step explanation:

Given you have collected 70 coins and stamps, and your friend has collected 1/5 as many coins and twice as many stamps, for a total of 95 coins and stamps, you want the numbers of each.

We can let x represent the number of coins you have collected. Then 70 -x is the number of stamps you have collected. The corresponding numbers for your friend are x/5 coins and 2(70-x) stamps. Your friend's collection is ...

  x/5 +2(70 -x) = 95

  -9/5x +140 = 95 . . . . . . . . simplify

  45 = 9/5x . . . . . . . . . . . . . add 9/5x -95

  25 = x . . . . . . . . . . . . . . . . multiply by 5/9

  70 -x = 45

  1/5x = 5

  2(70 -x) = 90

You collect 25 coins and 45 stamps; your friend collects 5 coins and 90 stamps.

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We can rewrite 3^6 as the product of two whole-number powers of 3 as: [tex]3^4 . 3^2.[/tex]

To rewrite 3^6 as the product of two whole-number powers of 3, we need to find two exponents that add up to 6.

A way to do this is to use the fact that 3^4 = 81, which is close to 3^6.

Specifically, we have:

3^6 = 3^4 * 3^2

We can check that this is true by using the rule for multiplying exponents that states that a^m * a^n = a^(m+n).

Applying this rule, we get:

= 3^4 * 3^2

= 3^(4+2)

= 3^6.

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The ages of people who can get a driver's license can be represented in three forms as follows;

x ≥ 16

x ≤ 85

16 ≤ x ≤ 85

The ages of people who cannot get a driver's license can be represented in three forms as follows;

x ≤ 16

x ≥ 85

16 ≥ x ≥ 85

In Mathematics and Geometry, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the inequality symbols;

Assuming the variable x denotes the ages of people who wishes to obtain a driver's license, a set of inequality which can be used to represent those who are eligible in three forms include;

x ≥ 16

x ≤ 85

16 ≤ x ≤ 85

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The coordinate of points K, L and M is the location of the points on a coordinate plane

Here, we have,

to determine the missing coordinates:

The given parameters are:

K = (10, )

L = ( ,10)

M = (30, )

The question has missing parameters.

So, I will assume that the line is a perfectly horizontal line.

This means that the y-coordinates of points K, L and M are equal.

The y-coordinate of point L is 10.

So, we have:

K = (10, 10)

L = ( ,10)

M = (30, 10)

Assume that point L is halfway points K and M, then we have:

K = (10, 10)

L = (20, 10)

M = (30, 10)

See attachment for the diagram of the coordinate plane showing the line

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Answer: see attached image

Step-by-step explanation:

since the distance between each number is split into fourths, that is the interval it is increasing.

5/2 simplified is 2 and 1/2. so you go 2 tick marks after the 2.

-2 and 3/4 goes 3/4 (3 tickmarks) after the -2.

The centerline=  459.9

The upper control limit=  852.55.

The lower control limit = 67.25.

We determine the mean to set up the standard limits:

Mean is gotten as :

(465 + 502 + 395 + 488 + 472 + 462 + 524 + 434 + 481 + 386) / 10

mean = 459.9

we then find the Standard Deviation:

Standard Deviation = √variance

variance = ((465-459.9)^2 + (502-459.9)^2 + (395-459.9)^2 + (488-459.9)^2 + (472-459.9)^2 + (462-459.9)^2 + (524-459.9)^2 + (434-459.9)^2 + (481-459.9)^2 + (386-459.9)^2) / 9

variance = 17119.1

Standard Deviation = √(17119.1)

Standard Deviation= 130.88

Hence

Centerline = 459.9

upper control limit  = 459.9 + 3 x (130.88) = 852.55

lower control limit. = 459.9 - 3 x (130.88) = 67.25

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Answer:

Maliks fruit salad will taste more melony

Step-by-step explanation:

To determine whose fruit salad will taste more "melony," we can calculate the percentage of melon in each fruit salad for Aisha and Malik.

For Aisha's fruit salad:

Total cups of fruit = 4 cups of melon + 1 cup of apple = 5 cups

Percentage of melon = (4 cups of melon / 5 cups of total fruit) x 100%

For Malik's fruit salad:

Total cups of fruit = 13 cups of melon + 3 cups of apple = 16 cups

Percentage of melon = (13 cups of melon / 16 cups of total fruit) x 100%

Now we can calculate the percentages:

Aisha's percentage of melon:

(4 cups of melon / 5 cups of total fruit) x 100% = 80%

Malik's percentage of melon:

(13 cups of melon / 16 cups of total fruit) x 100% ≈ 81.25%

Based on the calculations, Malik's fruit salad has a higher percentage of melon (81.25%) compared to Aisha's fruit salad (80%). Therefore, Malik's fruit salad is likely to taste more "melony" compared to Aisha's fruit salad.

The sample mean for the number of years worked is 12.9, and 35% of the employees in the sample worked for the company for at least 10 years.

To find the sample mean, we need to add up all the years worked and divide by the total number of employees:

8 + 13 + 15 + 3 + 13 + 28 + 4 + 12 + 4 + 26 + 29 + 3 + 10 + 3 + 17 + 13 + 15 + 15 + 23 + 13 = 258

So, the sample mean is 258/20 = 12.9 (rounded to the nearest tenth).

To find the percentage of employees who worked for at least 10 years, we need to count the number of employees who worked for 10 years or more.

From the data, we see that 7 employees worked for at least 10 years (Employee Number: 2, 3, 5, 10, 11, 15, and 19). So, 7 out of 20 employees worked for at least 10 years, which is 35%. Therefore, the answer is 35% (rounded to the nearest integer).

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Okay, let's go through this step-by-step:

1) sin(A) = opposite/adjacent = BC/AB = 18/24 = 3/4

2) cos(A) = adjacent/hypotenuse = AB/AC = 24/30 = 4/5

3) tan(A) = sin(A) / cos(A) = (3/4) / (4/5) = 3/4

4) sec(A) = 1/cos(A) = 1/4/5 = 5/4

5) csc(A) = 1/sin(A) = 1/3/4 = 12/3

In simple terms:

sin(A) = 3/4

cos(A) = 4/5

tan(A) = 3/4

sec(A) = 5/4

csc(A) = 12/3

Let me know if you need more details.

Answer- e^x-1-3

Step-by-step explanation: