Triangle PQR has its vertices at the following coordinates:

P(5,-4) Q(3,-5) R(1,-1)

Give the coordinates of the image triangle P'Q'R' after a 90° counterclockwise rotation about the origin.

The car will need 6 gallons of gasoline to travel 102 miles

Given that the car travels 442 miles on 26 gallons of gasoline.

Because the more the distance, the more the volume of gasoline used, this is a direct proportion.

So, we have:

It will need 6 gallons.

The matching for different kinds of telescopes is shown. (Refer to the matching shown below.)

A telescope is a tool used to view distant objects through the electromagnetic radiation that they emit, absorb, or reflect.

The term "telescope" now refers to a broad range of instruments capable of detecting different regions of the electromagnetic spectrum, and in some cases, other types of detectors, in addition to the original meaning of the word, which was only an optical instrument using lenses, curved mirrors, or a combination of both, to observe distant objects.

Refracting telescopes with glass lenses were the first practical telescopes ever made, and they were developed in the Netherlands at the start of the 17th century.

The matching according to the question:

Reflecting Telescopes: The Hubble Space Telescope, the largest telescope in the world, is currently the one most frequently used by professional astronomers.

Refracting Telescopes: In Galileo's telescopes, very large telescopes become "top-heavy," incoming light travels through glass, and the largest reflecting telescope in the world has a diameter of one meter.

Therefore, the matching for different kinds of telescopes is shown.

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Answer:

50%

Step-by-step explanation:

We need to find the percent change from 120 pounds to 180 pounds. We can find it as follows :

\(\%=\dfrac{180-120}{120}\times 100\\\\=\dfrac{60}{120}\times 100\\\\=50\%\)

So, the percent change is equal to 50%.

Answer:

3/4

Step-by-step explanation:

You can do defactoring of the given polynomial in simpler polynomials to get it as a  product of prime polynomials (assuming that it itself is not a prime polynomial)

The product of prime polynomial which forms the given polynomial is given by

Option B: \(4x^2(2x-3)(x+6)\)

Those polynomials with integer coefficients that cannot be factored further, with factors of lower degree and integer coefficients are called prime polynomial.

(it is necessary that no factors exists having their coefficients are still integers and they're of lower degree)

We can try defactoring the given polynomial in simpler polynomials.

The given polynomial is

\(8x^4 + 36x^3 - 72x^2\\\)

Factorizing it can be done as

\(\begin{aligned} 8x^4 + 36x^3 - 72x^2 &= 4x^2(2x^2 + 9x - 18)\\&= 4x^2(2x^2 + 12x - 3x - 18)\\&= 4x^2(2x(x+6) -3(x+60))\\&= 4x^2(x+6)(2x-3)\\&=4x^2(2x-3)(x+6)\\\end{aligned}\)

Thus,

The product of prime polynomial which forms the given polynomial is given by

Option B: \(4x^2(2x-3)(x+6)\)

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Answer:This is 69

Step by step explanation:67 +2 is 69

Answer:

1,1 2,2 x=y so it would be the same numbers for both points

Step-by-step explanation:

The answer is 237.55

Define Percentage

A ratio or figure stated as a fraction of 100 is called a percentage. Although the abbreviations "pct.", and occasionally "pc" are also used, the percent symbol, "%," is frequently used to indicate it. A % is a number without dimensions and without a standard measurement.

Given expression is,

125.8% of £188.83

Rewrite it as,

⇒ 125.8 / 100 * 188.83

⇒ 1.258 * 188.83

⇒ 237.54814

⇒ 237.55 (rounded to 2 Decimal places)

Hence, the answer is 237.55

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The simplified form is 67/10x + 9. Option B

From the information given, we have that;

We have the algebraic expression written as;

3(7/5x + 4) - 2(3/2 - 5/4x)

expand the bracket, we have;

21/5x + 12 - 6/2 + 10/4x

Now, collect like terms, we get;

21/5x + 10/4x + 12 - 6/2

Find the lowest common multiple, we have;

(84x+50x)/20 + (24-6)/2

subtract and add the value, we have;

134/20x + 18/2

Divide the common terms, we get;

67/10x + 9

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What is the simplified form of 3(7/5x + 4) - 2(3/2 - 5/4x)?

A. 75/4 x +8

B. 67/10x + 9

C. 66/10x + 2

D. 67/10x - 9

SOLUTION

Given the question in the question tab, the following are the solution steps to get the rental cost for each movie and each video game.

Step 1: Write the representation for the two rentals

Let m represents movies

Let v represent video games

Step 2: Write the statements in form of a mathematical equation

Step 3: Solve the equations above simultaneously using elimination method to get the values of m and v

Rental cost for each movies is $3.25

Rental cost for each video games $5.50

Answer:

Surveys allow researchers to collect a large amount of data in a relatively short period.

They are less expensive than many other data collection techniques.

Surveys can also be created quickly and administered easily.

Step-by-step explanation:

Hope this helps

The second solution, y2, to the given second-order differential equation is y2(t) = t³.

To find the second solution using the method of reduction of order, we assume that the second solution can be written as y2(t) = v(t) y1(t), where y1(t) is the known first solution (given as y1(t) = t²).

We substitute y2(t) = v(t) y1(t) into the differential equation:

t²v''(t) + 2tv'(t) - 4t²v'(t) - 4tv(t) + 6t²v(t) = 0.

Simplifying the equation, we get:

t²v''(t) - 2tv'(t) + 2tv'(t) - 4tv(t) + 6t²v(t) = 0,

t²v''(t) + (2t - 4t + 6t²)v(t) = 0,

t²v''(t) + (6t² - 2t - 4t)v(t) = 0,

t²v''(t) + (6t² - 6t)v(t) = 0.

Dividing the equation by t², we have:

v''(t) + (6 - 6/t)v(t) = 0.

Now, let u(t) = v'(t), then u'(t) = v''(t). Substituting these into the equation, we get:

u'(t) + (6 - 6/t)u(t) = 0.

This is a first-order linear ordinary differential equation. We can solve it by separation of variables:

du/u = (6 - 6/t)dt,

ln|u| = 6t - 6ln|t| + C1,

|u| = e^(6t-6ln|t|+C1),

|u| = e^(6t-6ln|t|) * e^C1.

Let C = ±e^C1, then |u| = C * e^(6t-6ln|t|).

Now, integrating u(t) = ±C * e^(6t-6ln|t|) dt, we get:

v(t) = ±C * ∫[e^(6t-6ln|t|)] dt.

Evaluating the integral, we find:

v(t) = ±C * t³.

Therefore, the second solution is y2(t) = t³, which forms a fundamental set with the given first solution y1(t) = t².

Using the method of reduction of order, we found the second solution y2(t) = t³ to the given second-order differential equation t²y''(t) - 4ty'(t) + 6y(t) = 0. The fundamental set {y1(t) = t², y2(t) = t³} satisfies the differential equation and allows us to express the general solution of the equation.

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If a coastline decreases by 3 cm each year for a total of 8 years. the integer that represents the total loss of coastline is: -24. The integer that can be use to represent the decrease is -3 and the expression that represent the total loss of coastline over 8 years is 8 (-3).

Given:

Decrease in coastline=-3

Total number of years=8 years

Hence,

a. Integer that represent the total loss of coastline is:

Integer=8(-3)

Integer=-24 (Total loss)

b. The integer that can be use to represent the decrease is:

Decrease in coastline= -3

c. The expression that represent the total loss of coastline over 8 years is:

Expression= 8(-3)

Therefore If a coastline decreases by 3 cm each year for a total of 8 years. the integer that represents the total loss of coastline is: -24. The integer that can be use to represent the decrease is -3 and the expression that represent the total loss of coastline over 8 years is 8 (-3).

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Answer:

60 miles :)

Step-by-step explanation:

Answer:

The perimeter of the rectangle would be 60 miles

Step-by-step explanation:

First you do 224 divided by 14. The answer to that would be 16. Next is 16x2. That would be 32. 14x2 = 28. 32 + 28 = 60.

Hope this helped out :)

The cost for the replacement for the window as required to be determined in the task content is; $1826.77.

It follows from the task content that the cost of replacing the window is required to be determined.

Since the triangular window mimics an equilateral triangle, it's area, A given side length, a is;

A = √3/4 • a².

Ultimately, the area of the window is;

A = √3/4 • 15²

A = 97.42785793 ft².

Since glass costs; 18.75 per square foot; it follows that the cost of replacing the window is; 97.42785793 ft² × 18.75 = $1826.77.

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a. The sampling method that I would use for the given scenario is stratified random sampling.

b. The sampling method that I would use for the given scenario is systematic sampling.

Stratified random sampling is the most effective way to get accurate results in this situation. The university wants to conduct a mail survey of its alumni to collect information about their current employment status to justify charging a higher tuition rate than other universities. Stratified random sampling divides the population into strata based on certain characteristics.

In this case, strata would be the alumni that have different degrees from the university. The information collected through this method will be more representative of the population than simple random sampling. It ensures that each stratum is represented accurately in the sample. It also minimizes the variability within each stratum. It is an effective way of ensuring that each stratum is properly represented.  

Systematic sampling involves selecting every kth element from a list of the population. In this situation, the Environmental Protection Agency (EPA) wants to inspect a sample of 1000 containers from the 40,000 containers known to be in landfills to determine if leakage from the containers has occurred. As the exact location of the containers is no longer known, systematic sampling can be used.

EPA can select every 40th container from the list of the population, as 40,000/1000 = 40. In this way, they will have a random sample of 1000 containers from the 40,000 containers. This sampling method is less time-consuming and more cost-effective than other sampling methods. It also ensures the sample is representative of the entire population.

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Answer:

-3 + 4 = 1

Step-by-step explanation:

Minus with minus makes a positive.

X = 40c + 39 = 40(4) + 39 = 199 is the required smallest positive integer.

The smallest positive integer which when divided by 4, 5, 8 and 9 leaves remainder 3, 4, 7 and 8, respectively is 539.

The given equation isx ≡ 3 (mod 4)x ≡ 4 (mod 5)x ≡ 7 (mod 8)x ≡ 8 (mod 9)

By the first condition,x = 4a + 3

where a is any positive integer.

Substituting in the second condition,4a + 3 ≡ 4 (mod 5)⇒ 4a ≡ 1 (mod 5)⇒ a ≡ 4 (mod 5)

Let a = 5b + 4 (where b is any positive integer)

Substituting this in x = 4a + 3, we get

x = 4(5b + 4) + 3 ⇒ x = 20b + 19

Now, substituting this in the third condition,20b + 19 ≡ 7 (mod 8)⇒ 4b ≡ 4 (mod 8)⇒ b ≡ 1 (mod 2)Let b = 2c + 1

Substituting this in x = 20b + 19,x = 20(2c + 1) + 19 ⇒ x = 40c + 39

Finally, substituting this in the fourth condition,40c + 39 ≡ 8 (mod 9)⇒ 4c ≡ 5 (mod 9)

We know that the modular inverse of 4 modulo 9 is 7.

Multiplying both sides by 7,7(4c) ≡ 7(5) (mod 9)⇒ 28c ≡ 35 (mod 9)⇒ c ≡ 4 (mod 9)

Hence,x = 40c + 39 = 40(4) + 39 = 199 is the required smallest positive integer.

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The indefinite integral of t ln(t+3) dt is t²/2 ln(t+3) - t²/4 + C.

To solve the indefinite integral of t ln(t+3) dt, we can use integration by parts, with u = ln(t+3) and dv = t dt.

Applying integration by parts, we have:

∫ t ln(t+3) dt = t ∫ ln(t+3) dt - ∫(d/dt[t] ∫ ln(t+3) dt) dt

The first integral can be solved using u-substitution. Let u = t+3, then du = dt, and we have:

∫ ln(t+3) dt = ∫ ln(u) du = u ln(u) - u + C = (t+3) ln(t+3) - (t+3) + C

The second integral can be simplified by applying the product rule and using the result of the first integral. We have:

(d/dt[t] ∫ ln(t+3) dt) = (t+3) / (t+3) - 1 = 2 / (t+3)

Substituting these results back into the integration by parts formula, we get:

∫ t ln(t+3) dt = t [(t+3) ln(t+3) - (t+3)] - 2t + C

Simplifying and factoring, we obtain:

∫ t ln(t+3) dt = t²/2 ln(t+3) - t²/4 + C

Therefore, the indefinite integral of t ln(t+3) dt is t²/2 ln(t+3) - t²/4 + C.

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Answer: A

Step-by-step explanation:

hope this helps

The value of the probability of choosing a green lollipop and a purple lollipop is, 8 / 45

We have to given that;

One bag contains 3 red lollipops and 2 green lollipops.

And, The other bag contains four purple lollipops and five blue lollipops.

Hence, The probability of choosing a green lollipop is,

P₁ = 2 / 5

And, The probability of choosing a purple lollipop is,

P₂ = 4 / 9

Thus, The value of the probability of choosing a green lollipop and a purple lollipop is,

P = P₁ ×  P₂

P = 2/5 × 4/9

P = 8/45

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Answer:

3x, 4x | 5, 3

Step-by-step explanation:

The radius of the pool is 9.01 m.

Given,

In the question:

A circular pool is surrounded by a brick walkway 3 m wide.

The area of the walk- way is 198 m^2.

To find the Radius of the pool.

Now, According to the question:

"Area of the circle bounded by the outside edge of the walkway" minus "area of the pool" = "area of the walkway".

Let R = Radius of the pool

Area of the circle bounded by the outside edge of the walkway is:

\(\pi\)(R +3)^2

Area of the pool is:

\(\pi R^2\)

Now, Our equation is:;

\(\pi\)(R +3)^2 - \(\pi R^2\) = 198

\(\pi\)((R+3)^2 - \(R^2\)) = 198

Open the inner bracket :

\(\pi\)(\(R^2+6R+9-R^2\)) = 198

\(\pi\)(6R +9) = 198

6R+9 = 198/\(\pi\)

6R = 198/\(\pi\) - 9

R = (198/\(\pi\) - 9)/6

R = (198/(3.14) - 9)/6

R = (63.057 - 9)/6

R = 54.057/6

R = 9.01 meters

Hence, The radius of the pool is 9.01 m.

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Answer:

0.125

Step-by-step explanation:

Becuase math

The angles for which tan() is positive are B. 3/4 and D. 3. Using the reference angle of each given angle, we can determine which quadrant it falls in and whether the tangent is positive or negative.

The tangent function is defined as the ratio of the opposite side to the adjacent side of a right triangle. When the angle is acute (less than 90 degrees), the tangent is positive if and only if the opposite side is positive and the adjacent side is negative. This occurs in the second quadrant and fourth quadrant of the unit circle.

Using the reference angle of each given angle, we can determine which quadrant it falls in and whether the tangent is positive or negative.

A. - is not an angle measure, so it cannot be evaluated.

B. 3/4: The reference angle is 1/4 of a full rotation, which falls in the second quadrant. Therefore, the tangent is positive.

C. -3/4: The reference angle is 1/4 of a full rotation, which falls in the second quadrant. Therefore, the tangent is positive.

D. 3: The reference angle is 3/4 of a full rotation, which falls in the first quadrant. Therefore, the tangent is positive.

E. 7/6: The reference angle is 1/6 of a full rotation, which falls in the third quadrant. Therefore, the tangent is negative.

In summary, the angles for which tan() is positive are B. 3/4 and D. 3.

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No , it is not possible to form a triangle with side lengths 3.5, 3.5, and 9 , because of the rule that sum of any two sides must be greater than other side. So, Option D is correct.

A triangle is a two-dimensional shape formed by three line segments connecting three points. In order to form a triangle, a necessary and sufficient condition is that the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality theorem.

If the sum of the lengths of any two sides is less than or equal to the length of the third side, then the three line segments will not be able to form a closed shape, and therefore cannot form a triangle.

In the case of the triangle with side lengths 3.5, 3.5, and 9, the sum of the lengths of the first two sides is 7, which is less than the length of the third side, 9. Hence, these three side lengths do not satisfy the triangle inequality theorem, and therefore cannot form a triangle.

In general, to determine the type of triangle that can be formed with three given side lengths, it is necessary to compare the lengths of the sides and check if any two sides are equal in length (isosceles), or if all three sides are equal in length (equilateral). If none of the sides are equal, then the triangle is called scalene.

In this case, since it is not possible to form a triangle with the given side lengths, it is not possible to determine the type of triangle that would be formed.

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Answer:

x = 55 degrees

Step-by-step explanation:

The sum of the internal angles of a triangle results in 180°

x + 50° + 75° = 180°

x + 125° =180°

x = 180° - 125°

x = 55°

A. The expression 25c2 evaluates to 300.

B. To evaluate the expression 25c2, we need to calculate the value of 25 multiplied by the binomial coefficient 2.

The binomial coefficient, denoted as "c" or sometimes represented by "C" or "choose," is a mathematical function that calculates the number of ways to choose a certain number of items from a larger set.

The binomial coefficient can be calculated using the formula:

nCk = n! / (k!(n-k)!)

In this case, we have 25C2, which means we need to calculate the number of ways to choose 2 items from a set of 25 items.

Plugging the values into the formula, we have:

25C2 = 25! / (2!(25-2)!)

     = 25! / (2! * 23!)

Calculating the factorials, we have:

25! = 25 * 24 * 23!

2! = 2 * 1

Substituting the values back into the equation, we get:

25C2 = (25 * 24 * 23!) / (2 * 1 * 23!)

Simplifying the expression, we find:

25C2 = 25 * 12

     = 300

Therefore, the expression 25c2 evaluates to 300 when expressed using the usual format for writing numbers.

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First and foremost, it is important to note that human skin color is determined by a variety of factors, including genetics, environmental influences, and geographic location. In fact, skin color is the result of the amount and type of melanin pigment produced by specialized cells in the skin called melanocytes.

There are two main types of melanin pigment: eumelanin and pheomelanin. Eumelanin is responsible for producing darker skin colors, while pheomelanin produces lighter skin colors. However, the amount and distribution of these pigments can vary greatly between individuals, resulting in a wide range of skin tones.

Additionally, skin color can also be influenced by factors such as sun exposure, age, and hormonal changes. For example, prolonged sun exposure can lead to darker skin, while aging and hormonal changes can cause changes in skin pigmentation. Furthermore, it is important to note that skin color is not always clearly defined or easily categorized into specific shades. There can be a great deal of variation within a single skin tone, as well as overlap between different skin tones. This is due to the complex interplay of genetic and environmental factors that contribute to skin color.

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