Find the partial sum sn of the geometric sequence that satisfies the given conditions. a3 = 36, a6 = 288, n = 5

Answer:

2.2 miles

Step-by-step explanation:

Coordinates of Joe's home = (2, 3)

Coordinates of the park = (1, 1)

The shortest distance = the distance between the two points

Use distance formula, \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) to calculate the shortest distance.

Where,

\( (2, 3) = (x_1, y_1) \)

\( (1, 1) = (x_2, y_2) \)

Plug in the values into the formula

\( d = \sqrt{(1 - 2)^2 + (1 - 3)^2} \)

\( d = \sqrt{(-1)^2 + (-2)^2} \)

\( d = \sqrt{1 + 4} \)

\( d = \sqrt{5} \)

\( d = 2.2 \) (nearest tenth)

Shortest distance between Joe's home and the park = 2.2 miles

The probability of randomly drawing a student from the whole population of fourth-graders that would meet the selection criterion for the gifted program is, 0.03.

What is probability?

Mathematical explanations of the likelihood that an event will occur or that a statement is true are referred to as probabilities. A number between 0 and 1 represents the likelihood of an event, with 0 generally denoting impossibility and 1 denoting certainty.

We have to find the probability of randomly drawing a student from the whole population of fourth-graders that would meet the selection criterion for the gifted program.

Think about the right-tailed region. Simply put, consider whether 97% of the pupils can be chosen for the talented programme.

Since we are interested in students who scored at or above 97%, the likelihood that a student will be chosen at random and match the selection criteria is 97/100 or 0.97.

Therefore, The probability of randomly drawing a student from the whole population of fourth-graders that would meet the selection criterion for the gifted program is, 0.03.

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a)  The x-intercepts of the graph of f(x) are x = 2.5 and x = -1.

b) The coordinates of the vertex are (0.75, -5.125).

c) By using the x-intercepts and vertex obtained in Parts A and B, we can accurately depict the shape and positioning of the parabolic graph of f(x).

Part A: To find the x-intercepts of the graph of f(x), we set f(x) equal to zero and solve for x.

Setting f(x) = 0:

\(2x^2 - 3x - 5 = 0\)

To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 2, b = -3, and c = -5. Substituting these values into the quadratic formula:

x = (-(-3) ± √((-3)^2 - 4(2)(-5))) / (2(2))

x = (3 ± √(9 + 40)) / 4

x = (3 ± √49) / 4

x = (3 ± 7) / 4

This gives us two possible solutions:

x1 = (3 + 7) / 4 = 10/4 = 2.5

x2 = (3 - 7) / 4 = -4/4 = -1

Therefore, the x-intercepts of the graph of f(x) are x = 2.5 and x = -1.

Part B: To determine whether the vertex of the graph of f(x) is a maximum or minimum, we need to consider the coefficient of the x^2 term in the function f(x). In this case, the coefficient is positive (2), which means the parabola opens upward and the vertex represents a minimum point.

To find the coordinates of the vertex, we can use the formula x = -b / (2a). In our equation, a = 2 and b = -3:

x = -(-3) / (2(2))

x = 3 / 4

x = 0.75

To find the corresponding y-coordinate, we substitute x = 0.75 into the function f(x):

f(0.75) = 2(0.75)^2 - 3(0.75) - 5

f(0.75) = 2(0.5625) - 2.25 - 5

f(0.75) = 1.125 - 2.25 - 5

f(0.75) = -5.125

Therefore, the coordinates of the vertex are (0.75, -5.125).

Part C: To graph the function f(x), we can follow these steps:

Plot the x-intercepts obtained in Part A: (2.5, 0) and (-1, 0).

Plot the vertex obtained in Part B: (0.75, -5.125).

Determine if the parabola opens upward (as determined in Part B) and draw a smooth curve passing through the points.

Extend the curve to the left and right of the vertex, ensuring symmetry.

Label the axes and any other relevant points or features.

By using the x-intercepts and vertex obtained in Parts A and B, we can accurately depict the shape and positioning of the parabolic graph of f(x).

The x-intercepts help determine where the graph intersects the x-axis, and the vertex helps establish the lowest point (minimum) of the parabola. The resulting graph should show a U-shaped curve opening upward with the vertex at (0.75, -5.125) and the x-intercepts at (2.5, 0) and (-1, 0).

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For the parabola y = 2x² + 3x + 1,

a. i.  The x-intercepts are x = -1/2 and x = -1

a. ii. The y-intercept is y = 1

b. i. The line of symmetry is at x = -3/4

b. ii. The equation of the line of symmetry is x = -3/4

c.  The vertex is at (-3/4, -1/8)

A parabola is part of a conic section which is a curve.

Given the parabola y = 2x² + 3x + 1, we find the x- intercept when y = 0.

So, y = 2x² + 3x + 1

2x² + 3x + 1 = 0

2x² + 2x + x + 1 = 0

Factorizing, we have

2x(x + 1) + (x + 1) = 0

(2x + 1)(x + 1) = 0

2x + 1 = 0 or x + 1 = 0

2x = -1 or x = -1

x = -1/2 or x = -1

So, the x-intercepts are x = -1/2 and x = -1

Given the parabola y = 2x² + 3x + 1, we find the y- intercept when x = 0.

So, y = 2x² + 3x + 1

So, y = 2(0)² + 3(0) + 1

y = 0 + 0 + 1

y = 1

So, the y-intercept is y = 1

The line of symmetry passes through the vertex of the parabola where dy/dx = 0.

So, y = 2x² + 3x + 1

dy/dx = d(2x² + 3x + 1)/dx

= d(2x²)/dx + d3x/dx + d1/dx

= 4x + 3 + 0

= 4x + 3

So, when dy/dx = 0,we have

4x + 3 = 0

4x = -3

x = -3/4

So, the line of symmetry is at x = -3/4

The equation of the line of symmetry is x = -3/4

Since the vertex of the parabola is a t x = -3/4, substituting this into the equation, we have

y = 2x² + 3x + 1

y = 2(-3/4)² + 3(-3/4) + 1

y = 2(9/16) - 9/4 + 1

y = 9/8 - 9/4 + 1

y = (9 - 18 + 8)/8

y = (17 - 18)/8

y = -1/8

So, the vertex is at (-3/4, -1/8)

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Asymptotes are lines that a curve approaches but does not intersect. A curve may have vertical, horizontal, or slant asymptotes or none at all. Asymptotes are critical features of the curve since they are used to help graph the function.

We will use the following steps to find asymptotes of the function h(x) = 4 + (5/x):

Vertical Asymptotes:Vertical asymptotes occur when the denominator is equal to zero. So for the function

h(x) = 4 + (5/x), the denominator x cannot be equal to 0.

Therefore x = 0 is the equation of the vertical asymptote.Horizontal Asymptotes:

Horizontal asymptotes occur when the value of y approaches a constant as x becomes extremely large.

For the function h(x) = 4 + (5/x), we need to take the limit as x approaches infinity and as x approaches negative infinity respectively.

(i) As x approaches infinity,y = 4 + 0= 4(ii) As x approaches negative infinity,y = 4 + 0= 4.

Therefore, the equation of the horizontal asymptote is y = 4.

Finally, the vertical asymptote equation is x = 0, and the horizontal asymptote equation is y = 4 for the function h(x) = 4 + (5/x).

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To find the derivatives of the given expressions, we can apply the chain rule, which states that the derivative of a composition of functions is equal to the derivative of the outer function multiplied by the derivative of the inner function.

(a) To find the derivative of f(g(x)), we start by differentiating the outer function f with respect to the inner function g(x), and then multiply it by the derivative of the inner function g(x) with respect to x.

df/dx = df/dg * dg/dx

df/dx = cos(g(x)) * (2x)

(b) To find the derivative of g(f(x)), we differentiate the outer function g with respect to the inner function f(x), and then multiply it by the derivative of the inner function f(x) with respect to x.

dg/dx = dg/df * df/dx

dg/dx = 2f(x) * cos(x)

Hence, the derivatives are:

(a) d/dx(f(g(x))) = cos(g(x)) * (2x)

(b) d/dx(g(f(x))) = 2f(x) * cos(x)

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Answer:

linear

Step-by-step explanation:

Answer: its a trinomial

Step-by-step explanation:

In order to decide the best study locations, the researchers must determine which study location is the most suitable for each student. This ensures that students are able to learn effectively and have the best chance of succeeding.

A randomized study location experiment will help researchers find out which study locations are best suited to each type of learner.

To decide study locations, the researcher should use an objective chance procedure, such as flipping a fair coin. The objective chance procedure will guarantee that each student has an equal chance of being assigned to a specific study location.

The researcher should ensure that both groups are matched as closely as possible in terms of characteristics that may be relevant to study success, such as ACT scores, GPAs, and socioeconomic backgrounds.

This guarantees that any variations in academic performance between the two groups are most likely due to differences in the study location and not to other external factors.

Therefore in order to decide the best study locations, the researchers must determine which study location is the most suitable for each student. This ensures that students are able to learn effectively and have the best chance of succeeding.

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Least to Greatest:

2/10, 2/5, 1/2, 5/3

Explanation:

They are all different fractions.

Hopefully, this helps! :D

Practice with the number line!

Answer: 3-dimensional prism

Step-by-step explanation:

Answer:

Answer:

These are the missing numbers

Sec1

A. 12, 29/2, 17 (arithmetic)

C. 3/16, 3/32, 3/64 (geometric)

D. 17, 33, 65 (neither)

Sec2

A. 8, 48 (growth)

B. 16, 32 (growth)

C. 1, 2 (rate)

Step-by-step explanation:

Section 1

A. add by 2.5 or 5/2

C. multiply by 1/2

D. double the value-added

Section2

A & B. multiply by 2

C. add 1/2

Answer:

First Equation:

Slope Intercept Form: y=-1/2x+6

Standard Form: -1/2x+6

Second Equation:

Slope Intercept Form: y=1x-3 OR y=x-3

Standard Form: 1x-3 OR x-3

Step-by-step explanation:

We can just solve the equation by using the inequalities method.

First Equation:

y - 4 = - 1/2(x - 4)

y - 4= - 1/2x + 2

y = - 1/2x + 6

Second Equation:

y + 7 = 1/3(3x + 12)

y + 7 = 1x + 4

y = 1x - 3

Hope this helped!

Answer:

lol bro it's your exam

Step-by-step explanation:

I can give u hint:

for example,if you know the width of the shape,

divide one width by the other to find the scale factor

I hope it will help you.

The pythagorean triple is at least one of the legs of a Pythagorean triple must be even.

The Pythagoras theorem states that the square of the longest side must be equal to the sum of the square of the other two sides in a right-angle triangle.

|AC|^2 = |AB|^2 + |BC|^2    

We are given that;

(2 y2)2 + (2xy)2 = (x² + y2)2

1. Let's choose 2 and 5 as our two numbers, and let 2 replace y, while 5 replaces x. Substituting these values into the Pythagorean identity, we get:

(2y)² + (2xy)² = (x² + y²)²

⇒ (2(2))² + (2(5)(2))² = (5² + 2²)²

⇒ 4² + 20² = 29²

2. We chose the number 2 to replace y because we want to create a Pythagorean triple, which means we need to find values of x, y, and z (the hypotenuse) that satisfy the Pythagorean theorem, a² + b² = c². In this case, since we chose 2 as y, we can think of this as the length of one of the legs of a right triangle, and we want to find values of x and z that make this true.

3. To find a Pythagorean triple using these numbers, we just need to set x = 5 and z = 29 (which we obtained from the Pythagorean identity). Then we have:

a = 2, b = 5, c = 29

and we can verify that this is a Pythagorean triple by checking that a² + b² = c²:

2² + 5² = 4 + 25 = 29 = 29²

4. At least one leg of the triangle that the Pythagorean triple represents must have an even-numbered length because in any Pythagorean triple, one of the legs must be even.  If a and b are both even, then their sum (c) will also be even. Therefore, at least one of the legs of a Pythagorean triple must be even. In our example, we chose 2 to be the even number, and indeed, it is the length of one of the legs of the right triangle represented by the Pythagorean triple.

Therefore, by pythagoras theorem the answer will be at least one of the legs of a Pythagorean triple must be even.

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The person that has the higher average speed is Teresa.

Average speed is calculated by dividing the total distance that something has travelled by the total amount of time it took it to travel that distance.

Therefore, Karl drove 40 miles in 50 minutes and Teresa drove 35 miles in 42 minutes.

The person that has the higher average speed can be calculated as follows;

average speed of Karl = distance / time

average speed of Karl = 40 / 50

average speed of Karl = 4 / 5 miles per minutes

average speed of Teresa = distance / time

average speed of Teresa = 35 / 42

average speed of Teresa = 5 / 6 miles per minutes

Therefore, Teresa had the higher average speed.

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Answer: it’s 20m long

Step-by-step explanation:

I can’t explain .—.

Answer:

20 meters

multiply the length by itself then multiply the width by itself then add them together then the square root of the total is the diagonal length.

step-by-step explanation :

16 x 16 = 256

12 x 12 = 144

256 + 144 = 400 sq rt = 20

Answer. The diagonal length is 20 meters

Answer:

5r+4

Step-by-step explanation:

Let's simplify step-by-step.

6r−r+4

=6r+−r+4

Combine Like Terms:

=6r+−r+4

=(6r+−r)+(4)

=5r+4

By the product rule of differentiation, the derivative of (x² - 3x + 2)(x + 2) with respect to x is: 2x² - 7x + 4.

d/dx [(x² - 3x + 2) (x + 2)]

= (x² - 3x + 2)'(x + 2) + (x² - 3x + 2) (x + 2)'

= (2x - 3) (x + 2) + (x² - 3x + 2) (1)

= 2x² - 7x + 4.

Differentiation is a concept in calculus that involves finding the rate of change or the slope of a function at a particular point. It involves finding the derivative of a function, which is a measure of how the function changes as its input changes. The derivative of a function can be used to determine important features of the function, such as maximum or minimum points, as well as to study the behavior of the function over an interval.

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Answer:

a) $2000

b) 15000(3.20)/(24+x) = new gasoline cost (replace with variable of your choice)

c) 2000-(15000(3.20)/(24+x)) = money saved (replace with variable)

d) ~34.29 miles (34.28571 miles)

Step-by-step explanation:

a) [15000(3.20)]/24 = 2000

b) use the formula above, but the denominator is 24+x (or variable of your choice) because the amount of mpg is increased

c) take the formula from a, then subtract formula in b from it

d) take formula c, replace the variable (variable = money saved) with 600

Answer:

answer is d

Step-by-step explanation:

I will se anywhere

The correct options that could be an example of a function with a domain (-∞0,00) and a range (-∞,4) are given below.Option A. V = -(0.25)x - 4 Option B. V = − (0.25)x+4

A function can be defined as a special relation where each input has exactly one output. The set of values that a function takes as input is known as the domain of the function. The set of all output values that are obtained by evaluating a function is known as the range of the function.

From the given options, only option A and option B are the functions that satisfy the condition.Both of the options are linear equations and graph of linear equation is always a straight line. By solving both of the given options, we will get the range as (-∞, 4) and domain as (-∞, 0).Hence, the correct options that could be an example of a function with a domain (-∞0,00) and a range (-∞,4) are option A and option B.

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The division gives the Quotient 180.

Division in mathematics is the process of dividing an amount into equal parts. For instance, we may split a group of 20 people into four groups of 5, five groups of 4, and so on.

Given:

We have to divide 1 thousand 8 hundreds divide by 10

So, 1800 ÷ 10

The Division is as follows:

                                        10 | 1800 | 180

                                               10

                                             ____

                                               80

                                               80

                                               ____

                                                  0

Thus, the Quotient is 180.

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Answer: area= 38, perimeter=34

Step-by-step explanation: a=lw,p= sum of all sides

The area of the biggest, middle and smallest shapes are 14 cm²,16 cm², and 8 cm² respectively while the perimeter of the biggest, middle and smallest shapes are 18 cm,16cm, and 12 cm respectively.

Perimeter is the addition of an external measure of sides it will always positive.

A perimeter is a length and has a degree of the unit as one.

Perimeter of square = side + side + side + side

The given object is divided into three sections.

Biggest section :

Area = 2 x 7 = 14 cm²

Perimeter = 2(7 + 2) = 18 cm

Middle section :

Area = 4 x 4 = 16cm²

Perimeter = 2(4 + 4) = 16cm

Smallest section :

Area = 2 x 4 = 8 cm²

Perimeter = 2(2 + 4) = 12 cm

Hence "The area of the biggest, middle and smallest shapes are 14 cm²,16 cm², and 8 cm² respectively while the perimeter of the biggest, middle and smallest shapes are 18 cm,16cm, and 12 cm respectively".

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The thickness of the clay layer, which drains only from the upper surface, can be determined based on the consolidation settlement observations. With 50% of consolidation settlement occurring in 1 hour for a 25 mm thick specimen, and a total primary consolidation settlement of 35 mm occurring over many years, the thickness of the clay layer is approximately 87.5 mm.

The consolidation settlement of a clay specimen can be used to estimate the thickness of a clay layer that drains only from the upper surface. In this case, the observed settlement data provides valuable information.

Firstly, we know that 50% of the consolidation settlement occurs in 1 hour for a 25 mm thick clay specimen. This is an important parameter for calculating the coefficient of consolidation (Cv) using Terzaghi's theory. From the Cv value, we can estimate the time required for full consolidation settlement.

Secondly, we are given that the same clay settles 10 mm over 10 years and eventually settles a total of 35 mm over a longer period. This long-term settlement is known as the total primary consolidation settlement. By comparing this settlement value with the settlement data from the oedometer test, we can determine the thickness of the clay layer.

To calculate the thickness, we can use the concept of the consolidation settlement ratio. The ratio of the total primary consolidation settlement to the consolidation settlement at 50% completion is equal to the ratio of the total thickness to the thickness at 50% completion. Applying this ratio, we can determine that the thickness of the clay layer, which drains only from the upper surface, is approximately 87.5 mm.

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Answer:

A. Reflection over the x-axis.

Step-by-step explanation:

Now, we present how to transform \(f(x) = x\) into \(g(x) = -5\cdot (x-3)-8\) below:

1) \(f(x) = x\) Given.

2) \(f(x-3) = x - 3\) Horizontal shift to the right.

3) \(-5\cdot [f(x-3)] = -5\cdot (x-3)\) Vertical scale factor.

4) \(-5\cdot f(x-3)-8 = -5\cdot (x-3)-8\) Vertical shift downwards.

5) \(g(x) = -5\cdot (x-3)-8\) Definition/Result

In consequence, the only transformation which was not done to the linear parent function was a reflection over the x-axis. The correct answer is A.

The value of angle S is 53°

Exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of two remote interior angles.

With this theorem we can say that

7x+2 = 4x+13+19

collecting like terms

7x -4x = 13+19-2

3x = 30

divide both sides by 3

x = 30/3

x = 10

Since x = 10

angle S = 4x+13

angle S = 4(10) +13

= 40+13

= 53°

Therefore the measure of angle S is 53°

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The equation in the form (x + a)² = b is (x + 2)² = 16

The equation is given as:

x² + 4x = 12​

Take the coefficient of x

k = 4

Divide by 2

k/2 = 2

Square both sides

(k/2)² = 4

Add this result to both sides of x² + 4x = 12​

x² + 4x + 4 = 12​ + 4

Evaluate the sum

x² + 4x + 4 = 16

Express as a perfect square

(x + 2)² = 16

Hence, the equation in the form (x + a)² = b is (x + 2)² = 16

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The total number of students who did not have a dog or a cat was 5

Jennifer conducted a survey about the pets of students in her class. In total, 27 students participated in the survey. Among these, 14 students had cats, 12 students had dogs, and 4 students had both cats and dogs.

Jennifer took a survey among the students in her class to know about their pets. The total number of students who took part in the survey was 27. She found that 14 students had cats, 12 students had dogs, and 4 students had both cats and dogs.

Now, we need to calculate the number of students who don't have a dog or a cat. Let's start by calculating the total number of students who had either a dog or a cat.There were 14 students with cats and 12 students with dogs, which means that 26 students had either a cat or a dog.

However, since 4 students had both a cat and a dog, we must subtract this from our total. Therefore, there were 22 students with either a cat or a dog.Now, we need to subtract this from the total number of students who took the survey. That is,27 - 22 = 5 students Therefore, there were 5 students who did not have a dog or a cat.

The total number of students who did not have a dog or a cat was 5.

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Answer:

The answer is B. Tyrone's unit rate is 1 more mile per hour then John’s unit rate.

Step-by-step explanation:

First of all at the bottom is what the graph and the answer options are.

Hope this helped! :D