please help me im doing pre alg and i cant figure out wether i shade up or to the side; graph x > 1

That hypotenuse is the radius of the second circle which is equal to 15.

The segment joining the centers of the circles bisects the common chord.

So in the circle with radius 13, the common chord, the segment joining the centers of the circles, and the radius to an endpoint of the common chord form a right triangle with hypotenuse 13 and one leg 12; that makes the distance from the center of that circle to the common chord 5.

Since the length of the segment joining the two circles is 14, the distance from the center of the other circle to the common chord is 14-5=9.

Then in that other circle, we have a right triangle with legs 9 and 12, making the hypotenuse 15.

And that hypotenuse is the radius of the second circle which is equal to 15.

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Answer: Your welcome!

Step-by-step explanation:

The new balance after the first minimum payment is made is $328.5. This is calculated by taking the balance of $350 and subtracting 7% of the balance, which is $24.50.

The minimum payment that is due the next month is $23.04. This is calculated by taking 7% of the new balance of $328.5.

Answer:

1) y² - 5y = 750

2) 750 -y(y -5) = 0

3) (y + 25)(y -30) = 0

Step-by-step explanation:

Area of rectangle = 750 ft²

                 length = y ft

                width = (y - 5) ft

Area of rectangle = 750

       length * width = 750

           y (y -5) = 750

          y*y - 5*y = 750

          y² - 5y = 750

                    0 = 750 - y(y -5)

          750 - y(y-5) = 0

        y² - 5y - 750 = 0

        Sum = -5

      Product = -750

      Factors = -30 , 25   {-30 + 25 = -5  &  (-30)*25 = -750}

y² - 30y + 25y - 750 = 0      {Rewrite the middle term using the factors}

y(y - 30) +25(y - 30) = 0

      (y - 30)(y + 25) = 0        

The correct answer is Option 3 (0,2). The medians of a triangle intersect at the midpoint of the opposite side of the triangle.

Triangle is a three-sided geometric shape with three angles and three vertices. Triangles can be classified into different types based on the lengths of their sides and the angles between them. The three most commonly referred to types include the equilateral, isosceles and scalene triangle. An equilateral triangle has three sides of equal length, while an isosceles triangle has two sides of equal length. A scalene triangle has no equal sides or angles. All three types of triangles have interior angles that add up to 180 degrees and all three sides must be connected. All triangles are two-dimensional shapes, meaning they have no thickness or depth.

In triangle ABC, the opposite side is the line segment BC, which has endpoints at (-3,2) and (1,2). The midpoint of this line segment is (0,2).

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The given sentence (exist t elementof R) (t middot x = 20) is an open sentence because it contains a variable x that is not specified. A statement, on the other hand, is a sentence that can be classified as either true or false, without any variables or unknowns.

If 5 is substituted for x, the resulting sentence is a statement: (exist t elementof R) (t middot 5 = 20). The statement is false because there is no real number t that can be multiplied by 5 to get 20.

If pi is substituted for x, the resulting sentence is a statement: (exist t elementof R) (t middot pi = 20). The statement is false because there is no real number t that can be multiplied by pi to get 20.

If 0 is substituted for x, the resulting sentence is a statement: (exist t elementof R) (t middot 0 = 20). The statement is false because any number multiplied by 0 is always 0, not 20.

The truth set of the open sentence (exist t elementof R) (t middot x = 20) is the set of all real numbers t such that their product with x is equal to 20. In other words, the truth set is {t elementof R | t middot x = 20}.

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The probability that the random variable X will be less than or equal to x. A uniform distribution's CDF is a linear function that increases from 0 to 1 across the range [a, b].

The uniform probability density function is given as follows:

If a = x = b = 0, then fx (x) = 1/(b-a), otherwise

where a and b are the uniform distribution's lower and upper bounds, respectively.

This means that the probability density function fx(x) is constant and equal to 1/(b-a) for any value of x between a and b, indicating that the probability of x taking a value in that interval is proportional to the width of the interval.

The probability density function fx (x) is zero outside of this range, It implies that the probability of x taking a value outside of [a, b] is zero. This is because the uniform distribution is only defined within the range [a, b].

It's the cumulative distribution function (CDF) used to express the probability density function for a uniform distribution:

Fx (x) = (x-a)/(b-a)

where a = x = b = 0, x a = 1, and x > b.

It shows the probability that the random variable X will be less than or equal to x. A uniform distribution's CDF is a linear function that increases from 0 to 1 across the range [a, b].

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The CPU (Central Processing Unit) performs all arithmetic operations (for example, addition and subtraction) and all logic operations (such as sorting and comparing numbers).

The acronym that fills in the blank is CPU, which stands for Central Processing Unit. The CPU is essentially the brain of a computer, responsible for processing all the information that flows through the system.

Arithmetic operations are those that involve mathematical calculations, such as addition, subtraction, multiplication, and division. These are relatively straightforward tasks for the CPU, which can perform them quickly and accurately.

Logic operations, on the other hand, involve comparing and manipulating data based on various conditions.

CPU has the ability to perform logic operations is a crucial aspect of computing, as it allows computers to make decisions and process information based on a set of rules or conditions.

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Answer:

[tex]\dfrac{4a^6b^5}{c^3}[/tex]

Step-by-step explanation:

Given expression:

[tex]\dfrac{8a^4b^{-3}c^{12}}{2a^{-2}b^{-8}c^{15}}[/tex]

Separate like terms:

[tex]\dfrac{8}{2} \cdot \dfrac{a^4}{a^{-2}}\cdot \dfrac{b^{-3}}{b^{-8}} \cdot \dfrac{c^{12}}{c^{15}}[/tex]

Divide the numbers:

[tex]4\cdot \dfrac{a^4}{a^{-2}}\cdot \dfrac{b^{-3}}{b^{-8}} \cdot \dfrac{c^{12}}{c^{15}}[/tex]

[tex]\textsf{Apply the exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}:[/tex]

[tex]4 \cdot a^{4-(-2)} \cdot b^{-3-(-8)} \cdot c^{12-15}[/tex]

Simplify the exponents:

[tex]4 \cdot a^6 \cdot b^5 \cdot c^{-3}[/tex]

[tex]\textsf{Apply the exponent rule} \quad a^{-n}=\dfrac{1}{a^n}:[/tex]

[tex]4 \cdot a^6 \cdot b^5 \cdot \dfrac{1}{c^3}[/tex]

Simplify:

[tex]\dfrac{4a^6b^5}{c^3}[/tex]

Answer:

$1400 divided by 10 = $140

$1400 - $140 = $1260

Answer:Yes, they are all correct

Step-by-step explanation:

From the attached image, we see that the sign is showing;

Parking lot: ¾ miles

Summit: 1½ miles

Summit distance can also be expressed as an improper fraction = 3/2 miles

Now, since they started hiking from the park to the summit, it means that they had moved ¾ mile from the parking lot and had 3/2 miles left to get to the summit.

Thus, total distance from parking lot to summit = ¾ + 3/2 = 9/4 miles

Now, let's analyze each of their statements;

Mai said they are a third of their way there.

They had covered 3/4 mile and the total distance is 9/4 miles.

Thus, fraction of total distance covered is;

9/4 ÷ 3/4 = 1/3.

So, Mai is correct

Clare said they have to go twice as far as they had already gone.

They had covered 3/4 miles.

Therefore, twice this = 2 × 3/4 = 6/4 = 3/2 which is same as the sign distance left to the summit.

Thus, Clare is correct

Tyler said that the total hike is three times as long as what we have already gone.

They had gone 3/4 miles

3 times this = 3 × 3/4 = 9/4 miles.

This tallies with the total distance calculated earlier.

Thus, they are all correct

The measure of side ML of the triangle MLK is ML = 6 units

If two triangles' corresponding angles are congruent and their corresponding sides are proportional, they are said to be similar triangles. In other words, similar triangles have the same shape but may or may not be the same size. The triangles are congruent if their corresponding sides are also of identical length.

Corresponding sides of similar triangles are in the same ratio. The ratio of area of similar triangles is the same as the ratio of the square of any pair of their corresponding sides

Given data ,

Let the first triangle be represented as ΔMLK

Let the second triangle be represented as ΔHJI

The measure of side HJ = 20 units

The measure of side HI = 36 units

The measure of side JI = 24 units

And ,

The measure of side LK = 5 units

The measure of side MK = 9 units

The measure of side ML = A units

Now , the triangles are similar

So , the corresponding sides of similar triangles are in the same ratio

The measure of side MK / measure of side HI = The measure of side ML / measure of side JI

Substituting the values in the equation , we get

A / 24 = 9 / 36

Multiply by 24 on both sides of the equation , we get

A = ( 24 x 9 ) / 36

A = 24 / 4

A = 6 units

Hence , the measure of side ML of triangle is 6 units

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The probability that the arrow will land on a section labeled S or T is 7/8.

Probability is a branch of mathematics that deals with the likelihood of an event occurring. It is the measure of the likelihood of an event occurring divided by the number of possible outcomes. Probability is used to determine the chances of a particular outcome occurring and can range from 0 to 1.

Therefore, it is likely that the arrow will land on one of these sections around 56 to 57 times out of 75 spins. It is impossible to predict exactly how many times the arrow will land on S or T, as this is a probability-based outcome. Generally speaking, with such a large number of spins, the result should be close to the probability of 7/8.

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The value of x is 24. The angle measures of each vertex of the yard are m∠A = 160°, m∠B = 142°, m∠C = 120°, m∠D = 156°, m∠E = 31°, m∠F = 111°.

What is a polygon?

A polygon is a closed polygonal chain made up of a finite number of straight line segments and is a type of plane figure in geometry. A polygon is a region that is bounded by a bounding circuit, a bounding plane, or both. A polygonal circuit's segments are referred to as its edges or sides.

The formula of the sum of interior angles of a polygon is (n - 2)×180°, where n is number of sides.

The number of sides of the given irregularly shaped garden is 6.

The sum of the interior angles of the garden is (6 - 2)×180° = 720°

Also given:

m∠A = (7x - 8)°, m∠B = (4x + 46)°, m∠C = (5x)° , m∠D = (6x + 12)°, m∠E = (x + 7)°, m∠F = (5x - 9)°.

The sum of the interior angles of the garden is

(7x - 8)° + (4x + 46)° + (5x)° + (6x + 12)° +  (x + 7)° + (5x - 9)°

Therefore,

(7x - 8)° + (4x + 46)° + (5x)° + (6x + 12)° +  (x + 7)° + (5x - 9)° = 720°

Combine like terms:

(7x +4x + 5x + 6x + x + 5x) + (-8 +46 + 12 + 7 - 9) = 720

28x + 48 = 720

Subtract 48 from both sides:

28x = 672

Divide both sides by 28:

x = 24

Putting x = 28 in the given angles:

m∠A = (7×24 - 8)° = 160°

m∠B = (4×24 + 46)°= 142°

m∠C = (5×24)° = 120°

m∠D = (6×24 + 12)°= 156°

m∠E = (24 + 7)° = 31°  

m∠F = (5×24 - 9)° = 111°

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In a survery of hospital employees here it is parameter or  statistic  data is given  in Question

Parameters are numeric values ​​that characterize the population as a whole. A statistic is a number that describes the characteristics of a sample.

For example, median income in the United States is a demographic parameter. Conversely, the median income for the US sample is a sample statistic. Both values ​​represent median income, but one is a parameter to the statistic. Easy to remember parameters and stats! Both are summary values ​​that describe the group and have a convenient mnemonic device to remind you which group is being described. Notice their initials.

Parameter = Population

Statistics = Sample

A population is the entire group of people, things, animals, trades, etc. that are being studied. A sample is part of a population.

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The number of muffins in each bag is 6

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

5 boxes of muffins with m number in each box equals 30​

Using the above as a guide, we have the following equation

5m = 30

Divide both sides by 5

m = 6

Hence, there are 6 muffins in each bag

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Complete question

5 boxes of muffins with m number in each box equals 30​

What is the number of muffin in each box

The correct expression for P(X = k) will be [tex]P(X = k) = \binom{k-1}{r-1} \left(\frac{1}{2}\right)^k[/tex]

Given,

The geometric distribution arises as the distribution of the number of times we flip a coin until it comes up heads. consider now the distribution of the number of flips until the -th head appears, where each coin flip comes up heads independently with probability.

The negative binomial distribution describes the probability of getting the -th success (in this case, heads) on the th trial, given a probability p of success on each trial.

We can write this as:

[tex]P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}[/tex]

where k is the number of flips needed to get r successes, including the rth success.

In this case, we want the distribution of the number of flips until the rth head appears,

so we can set p = 1/2,

since the coin is fair.

We also need to adjust the formula to count the number of flips, including the flip with the rth head.

Since we need r heads to appear, the last head must appear on the rth flip, so we can write:

[tex]P(X = k) = \binom{k-1}{r-1} \left(\frac{1}{2}\right)^r \left(\frac{1}{2}\right)^{k-r+1}[/tex]

Simplifying the expression,

we get:

[tex]P(X = k) = \binom{k-1}{r-1} \left(\frac{1}{2}\right)^k[/tex]

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a. The population density of Monaco is 19,307 people per square kilometer.

b. The population density of Vatican City is 37 people per square kilometer.

c. The Vatican City has the highest population density

What is population density?

Population density is measured by dividing the total area of a region in question by the total number of people that live there.

a. To find the population density of Monaco, we need to divide the population by the area, and then convert acres to square kilometers (since Vatican City's area is given in square kilometers).

499 acres x 0.00404686 square kilometers/acre = 2.02 square kilometers

The population density of Monaco is therefore:

39,000 people / 2.02 square kilometers = 19,307 people per square kilometer.

Rounded to the nearest whole number, the population density of Monaco is 19,307 people per square kilometer.

b. To find the population density of Vatican City, we first need to convert its area to acres:

0.44 square kilometers x 247.105 acres/square kilometer = 108.7 acres

The population density of Vatican City is then:

1,000 people / 108.7 acres = 9.2 people per acre

To compare with Monaco's population density, we need to convert this to people per square kilometer:

9.2 people per acre x 0.00404686 square kilometers/acre = 37.3 people per square kilometer

Rounded to the nearest whole number, the population density of Vatican City is 37 people per square kilometer.

c. Therefore, Vatican City has the highest population density, with a density of 37 people per square kilometer compared to Monaco's density of 19,307 people per square kilometer.

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Answer:

[tex]x=8.3[/tex]

The first option listed

Step-by-step explanation:

We can use the cosine function to evaluate [tex]x[/tex].

The definition of the cosine function is

[tex]\cos \theta=\frac{A}{H}[/tex]

Note

[tex]\theta[/tex] is the angle

[tex]A[/tex] is the side adjacent to the angle

[tex]H[/tex] is the hypotenuse

In this example we are given the hypotenuse and the angle.

Knowing these 2 values we can evaluate the adjacent side ([tex]x[/tex]).

Lets solve for [tex]A[/tex].

[tex]\cos \theta=\frac{A}{H}[/tex]

Multiplying both sides by [tex]H[/tex] lets us isolate [tex]A[/tex] ([tex]x[/tex]).

[tex]A=H*\cos \theta[/tex]

Numerical Evaluation

We are given

[tex]\theta = 41\textdegree\\H=11[/tex]

Inserting those values into our equation for [tex]A[/tex] ([tex]x[/tex]) yields

[tex]A=11*\cos 41[/tex]

[tex]A=8.30180534[/tex]

Rounding to the nearest tenth gives us

[tex]A=8.3[/tex]

[tex]x=8.3[/tex]

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If the production is increased by 10%, the new production levels for sport utility vehicles and pickup trucks are:

Factory 1: 110 sport utility vehicles

Factory 2: 99 sport utility vehicles

Factory 3: 77 sport utility vehicles, 66 pickup trucks

Factory 4: 33 sport utility vehicles, 66 pickup trucks

To find the production levels if production is increased by 10%, we need to multiply each entry of the matrix by 1.1:

A' = 1.1A = | 110 99 77 33 || 44 22 66 66 |

The matrix A' represents the new production levels. The entry in the first row and first column of A' (110) represents the number of sport utility vehicles produced at the first factory after the 10% increase in production. Similarly, the entry in the second row and third column of A' (66) represents the number of pickup trucks produced at the third factory after the 10% increase in production.

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The six trigonometric functions of theta are: sine (sinθ) = -7/3, cosine (cosθ) = -3/3, tangent (tanθ) = -7/3, cotangent (cotθ) = -3/7, secant (secθ) = -3/7, and cosecant (cscθ) = -7/3.

sinθ = -7/3

cosθ = -3/3

tanθ = -7/3

cotθ = -3/7

secθ = -3/7

cscθ = -7/3

The point (3,-7) is located on the terminal side of an angle θ in standard position, thus allowing us to calculate the six trigonometric functions of theta. The six trigonometric functions of θ are sine (sinθ), cosine (cosθ), tangent (tanθ), cotangent (cotθ), secant (secθ), and cosecant (cscθ). To calculate each of these functions, we must first calculate the ratio of the vertical and horizontal components of the point. In this case, the vertical component is -7 and the horizontal component is 3. Therefore, the ratio of these two components is -7/3. This ratio is equal to the value of sinθ, tanθ, and cscθ. To calculate the cosθ, cotθ, and secθ, we must take the reciprocal of the ratio of the vertical and horizontal components. In this case, the reciprocal is -3/7. Therefore, the exact values of the six trigonometric functions of theta are sinθ = -7/3, cosθ = -3/3, tanθ = -7/3, cotθ = -3/7, secθ = -3/7, and cscθ = -7/3.

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Answer:

Step-by-step explanation:

250 per week base

15%of 1800 he made is = 270

270+250 =520

He made 520$ in the week he made 1800$ in sales

Jaxson's salary is a combination of a fixed $250 plus 15% commission on his weekly sales. For $1800 in sales, he will earn $520. If the sales amount is x dollars, then he will earn $250 + 0.15*x dollars in a week.

Jaxson's income consists of a guaranteed base salary of $250 per week and a commission of 15% on his total weekly sales. If Jaxson made $1800 in sales in a week, his total income for the week would be his base salary plus his commission i.e., $250 + 0.15*$1800 = $520.

Similarly, if Jaxson made x dollars in sales in a week, his total income for that week would be his base salary plus his commission on the sales i.e., $250 + 0.15*x dollars.

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Leasing and buying are two popular options for acquiring a car or other assets. Both options have their own advantages and disadvantages, and the decision between leasing and buying will depend on individual needs and preferences. Below are some pros and cons of leasing and buying.

Initial and Monthly Payments: When it comes to initial and monthly payments, leasing typically requires lower upfront costs and lower monthly payments compared to buying. This is because leasing only covers the cost of the car's depreciation during the lease term, while buying involves financing the full cost of the vehicle.

Total Cost: When considering the total cost of leasing and buying, it's important to look beyond just the initial and monthly payments. While leasing may have lower monthly payments, the total cost of the lease over the long term may be higher than buying. This is because at the end of the lease term, the lessee has no equity in the car and must either lease a new car or purchase a new car.

Ownership and Customization: One of the main advantages of buying is that the car is owned outright, giving the owner the freedom to customize the vehicle as desired. With leasing, the lessee is limited to the terms of the lease agreement and cannot make any major modifications to the car.

Flexibility: Leasing provides greater flexibility in terms of upgrading to a newer car at the end of the lease term, while buying requires selling or trading in the old car to acquire a new one.

In summary, leasing and buying both have their own advantages and disadvantages, and the decision between the two will depend on individual needs and preferences.

Leasing may be a good option for those who prioritize lower monthly payments and want the flexibility to upgrade to a newer car every few years, while buying may be a good option for those who prioritize ownership, customization, and long-term cost savings.

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Answer:

Step-by-step explanation:

The set {0,1,2,3,4,5,6,7,8,9} has 2^10 = 1024 subsets. To see why, note that each element can be either in or out of a given subset, so there are 2 choices for each of the 10 elements, giving 2x2x2x2x2x2x2x2x2x2 = 1024 possible subsets.

a) The number of ways to deal the cards without getting one of the designated cards are equals to the 2250829575120.

b) The number of ways to deal each player 7 cards, regardless of whether the designated cards come out are equals to the 21945588357420.

c) The probability of a successful hand that will go all the way till everyone gets 7 cards is equals to the 0.1025.

Five friends group are playing poker one night. They have a standard 52-card deck. So, here total possible outcomes

= 52

Now, the designated cards are 6 of Spades, Jack of Diamonds. So,

a) Number of cards are in the deck that are not one of the designated cards = 52 - 2 = 50

Number of cards that need to be dealt in order for each player to have 7 cards

= 5× 7 = 35

Thus total possible number of ways

= ⁵⁰C₃₅ = 2250829575120, which are ways to deal the cards without getting one of the designated cards.

b) Number of cards are in the deck = 52

Number of cards that need to be dealt in order for each player to have 7 cards

= 5× 7 = 35

Thus total possible number of ways

= ⁵²C₃₅ = 21945588357420

Which are ways to deal each player 7 cards, regardless of whether the designated cards come out.

c) The probability of a successful hand that will go all the way till everyone gets 7 cards is = Number of ways to deal the cards without getting one of the designated cards/Total number of ays to deal the cards

= 2250829575120/21945588357420

= 0.10256410256

Hence, required probability is 0.10256.

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Complete question:

A group of 5 friends are playing poker one night, and one of the friends decides to try out a new game. They are using a standard 52-card deck. The dealer is going to deal the cards face up. There will be a round of betting after everyone gets one card. Another round of betting after each player gets a second card, etc. Once a total of 7 cards have been dealt to each player, the player with the best hand will win. However, if any player is dealt one of the designated cards, the dealer collects all cards, shuffles, and starts over. The designated cards are: 6 of Spades, Jack of Diamonds. The players wish to determine the likelihood of actually getting to play a hand without mucking the cards and starting over.

a) In how many ways can you deal the cards WITHOUT getting one of the designated cards? (Hint: Consider how may cards are in the deck that are NOT one of the designated cards and consider how many cards need to be dealt in order for each player to have 7 cards.)

b) In how many ways can you deal each player 7 cards, regardless of whether the designated cards come out?

c) What is the probability of a successful hand that will go all the way till everyone gets 7 cards? (Round your answer to 4 decimal places.) Recall, while using your calculator, that E10 means to move the decimal place 10 places to the right.

The surface area of the pyramid is 119 inches².

The area of a three dimensional object on it's outer surface is called the surface area of the object.

A square pyramid has a square base and 4 triangular faces.

Surface area of the pyramid is the base area plus 4 times the area of each triangular face.

Base is a square.

Base area = a², where a is the side length of the square.

Base edge of square = 7 inches

Base area = 7² = 49 inches²

Slant height of triangle = 5 inches

Area of a triangular face = [tex]\frac{1}{2}[/tex] × base × height

                                         = [tex]\frac{1}{2}[/tex] × 7 × 5

                                         = 17.5 inches²

Area of 4 triangular faces = 4 × 17.5 inches²

                                           = 70 inches²

Surface area of the pyramid = 49 inches² + 70 inches² = 119 inches²

Hence the total surface area is 119 inches².

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Your question is incomplete. The complete question is probably the one given below.

Find the surface area of the pyramid. The side lengths of the base are equal. Square pyramid with base edge of 7 inches, and a slant height measuring 5 inches, on a triangular face.

the correct option is A. 333 cm (tiny 3).

Radius is a term used in geometry to refer to the distance from the center of a circle, sphere, or other curved shape to its edge or surface. It is a key measurement used to calculate the area, circumference, volume, and surface area of these shapes. The radius is often represented by the symbol "r" and can be calculated using various formulas depending on the shape in question. In general, the radius is half the diameter of a circle or sphere, which is the distance across the shape through its center.

Given by the question.

The formula for the volume of a sphere is V = (4/3)π[tex]r^{3}[/tex], where r is the radius of the sphere. Since the diameter of the sphere is given, we can find the radius by dividing it by 2:

radius = diameter/2 = 8.6 cm/2 = 4.3 cm

Now we can plug this value into the volume. and solve for V:

V = (4/3)π[tex](4.3cm)^{3}[/tex]≈ 333.06 [tex]cm^{3}[/tex]

To the nearest whole, the volume is approximately 333. [tex]cm^{3}[/tex].

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If a is negative, the parabola opens down. Option C is the correct option.

What is a parabola?

Any point on a parabola is at an equal distance from both the focus, a fixed point, and the directrix, a fixed straight line. A parabola is a U-shaped plane curve. The topic of conic sections includes parabola, and all of its concepts are covered here.

A polynomial function of the second degree is a quadratic function. A quadratic function has this general form:

y = ax² + bx + c

A parabola is a name for a quadratic function's graph.

A parabola resembles the letter "U" or an upside-down "U" in general shape.

If the leading coefficient is greater than zero, the parabola opens upward; if the leading coefficient is less than zero, the parabola opens downward. This simple rule can be used to determine whether the graph of a quadratic function opens upward or downward.

Given equation is

y = ax²

The leading coefficient is a and it is a negative number. Thus the parabola opens down.

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Answer:

2:5

Step-by-step explanation:

We know

There are 9 girls and 6 boys taking golf lessons.

So, there are a total of 15 taking a golf lesson.

Write the ratio that compares the number of girls taking golf lessons to the total number of students taking golf lessons.

The ratio is

6:15 = 2:5

So, the ratio is  2:5

The equation of the function of the given graph is y = (x - 5)² + 3 shown.

The parabola equation into the vertex form:

(y-k) = a(x-h)²

Where (h,k) is the x and y-coordinates of the vertex.

According to the given graph, we have data as follows:

Points on the x and y-axis = (8, 12).

Vertex (h, k)= (5, 3).

Substitute the values of h = 5 and k = 3 in the above equation

y - 3 = a(x - 5)²

y = a(x - 5)² + 3

If the parabola contains the point (0, 0)

Substitute the point (8, 12) in the above equation

12 = a((8-5)² + 3

12 = a(3)² + 3

12 - 3 = 9a

9a = 9

a = 1

4a = -3

a = -3/4

So, the equation becomes y = (x - 5)² + 3

Hence, the required equation is y = (x - 5)² + 3 which represents the given parabola.

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The position vector of a particle that has an acceleration, a(t) = 19t i + eᵗ j + e⁻ᵗ k, is equals to the 19 (t³/6)i + (eᵗ - t )j - (e⁻ᵗ - 2t - 2)k.

We have, The acceleration vector function of a particle is defined as, a(t)

= 19t i + eᵗ j + e⁻ᵗ k and intial velocity and position is, v(0) = k, r(0) = j + k.

We have to calculate the position vector of a particle. Now, as we know, the acceleration of a particle is equals to derivative of velocity of particle with time. In other words, velocity is integration of acceleration with respect to time.

Mathematically, v(t) = ∫a(t)dt , let C be integration constant ( vector).

v(t) = ∫a(t)dt = ∫[ (19t) i + (eᵗ) j + (e⁻ᵗ) k] dt

=> v(t) = 19(t²/2) i + eᵗ j - e⁻ᵗ k + C

At t = 0 , v(0) = k

=> k = 19(0²/2) i + e⁰j - e⁻⁰ k + C

=> k = 0 + j - k + C

=> C = 2k - j

so, v(t) = 19(t²/2) i + eᵗ j - e⁻ᵗ k + 2k - j

= 19(t²/2) i + (eᵗ - 1) j - (e⁻ᵗ - 2) k

Now, Position of a particle is determined by integrating the velocity of particle with respect to time, r(t) = ∫v(t)dt , let D be integration constant ( vector). So, r(t)

= ∫[19(t²/2) i + (eᵗ - 1) j - (e⁻ᵗ - 2) k ] dt

= 19 (t³/6) i + eᵗ j - t j - e⁻ᵗ k + 2t k + D

At t = 0, r(0) = j + k

=> j + k = 19 (0³/6) i +e⁰ j - 0j - e⁻⁰ k +2× 0k +D

=> j + k = 0 + j - k + D

=> D = 2k

so, r(t) = 19 (t³/6) i + eᵗ j - t j - e⁻ᵗ k + 2t k + 2k

= 19 (t³/6)i + (eᵗ - t )j - (e⁻ᵗ - 2t - 2)k

Hence, required position vector is 19 (t³/6)i + (eᵗ - t )j - (e⁻ᵗ - 2t - 2)k.

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