The sum of the first 12 terms of an arithmetic progression is 156. What is the sum of the first and twelfth terms?

Check the picture below.

so then, the distance from the center to that point is really the radius.

[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{3}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{9}~,~\stackrel{y_2}{3})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ \stackrel{radius}{r}=\sqrt{(9-3)^2+(3-2)^2}\implies r=\sqrt{36+1}\implies r=\sqrt{37} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf \textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \qquad center~~(\stackrel{3}{ h},\stackrel{2}{ k})\qquad \qquad radius=\stackrel{\sqrt{37}}{ r} \\\\\\ (x-3)^2+(y-2)^2=(\sqrt{37})^2\implies (x-3)^2+(y-2)^2=37[/tex]

Using technology, the regression equation is

y=9.08292(1.09965)ˣ. 

The number of recommendations after 60 days is 2113.277 ≈ 2113.

Explanation

I used Desmos to graph the points, then entered the expression y₁~a*bˣ to calculate the regression.

I then substituted 60 in for x in the regression equation to get the answer of 2113.

Answer:

The weight of the enlarged paperweight is [tex]1.2\ lb[/tex]

Step-by-step explanation:

we know that

If two figures are similar, then the ratio of its volumes (or its weights) is equal to the scale factor elevated to the cube

Let

z-----> the scale factor

x ----> the enlarged paperweight weight

y ----> the original paperweight weight

[tex]z^{3}=\frac{x}{y}[/tex]

we have

[tex]z=2[/tex]

[tex]y=0.15\ lb[/tex]

substitute and solve for x

[tex]2^{3}=\frac{x}{0.15}[/tex]

[tex]x=2^{3}(0.15)=1.2\ lb[/tex]

Answer:

The product is equal to [tex]3\sqrt{15}[/tex]

Step-by-step explanation:

Ok,

First, you have

[tex](3\sqrt{-3})(\sqrt{-5})[/tex]

We must follow the rule for multiplying radicals, in this case:

[tex](3\sqrt[2]{-3})(\sqrt[2]{-5})=(3\sqrt[2]{(-3)(-5)})[/tex]

Note that the types of root, n, have to match, in this case is 2 for each root

Then, we know that two negative numbers multiplied give a positive number, in this case

[tex](3\sqrt[2]{-3} )(\sqrt[2]{-5})=(3\sqrt[2]{(-3)(-5)})=(3\sqrt[2]{(15)})[/tex]

So, the correct answer is [tex](3\sqrt{-3})(\sqrt{-5} )=3\sqrt{15}[/tex]

Answer:

There are a total of 4 + r + 6 marbles in the bag

The probability of a blue is  [tex]\frac{4}{10+r}[/tex]  

The probability of a red is  [tex]\frac{r}{10+r}[/tex]  

The probability of choosing a blue, replacing it and then a red is

[tex]\frac{4}{10+r}[/tex] × [tex]\frac{r}{10+r}[/tex]  =  [tex]\frac{4r}{100+20r+ r^{2} /[tex]

Step-by-step explanation:

Answer:

18 Quarts

Step-by-step explanation:

there are 4 quarts per gallon

I multiplied 4 x 5 = 20 then subtracted the 2 quarts left which comes out to 18 quarts used.

Answer:

The measure of angle 6 is m∠6=50°

Step-by-step explanation:

we know that

m∠3+m∠6=180° ----> supplementary angles (interior consecutive angles)

so

we have

m∠3=130°

substitute

130°+m∠6=180°

m∠6=180°-130°=50°

Answer:

  2³² -1

Step-by-step explanation:

The sequence has a₁ = 1 and r = 2. Filling in the numbers for n=32, we have ...

S₃₂ = 1·(1 -2³²)/(1 -2)

S₃₂ = 2³² -1

Answer:

$48

Step-by-step explanation:

we know that

$80 represent 100%

so

using proportion

Find how much is 60%

80/100=x/60

x=80*60/100

x=$48

Answer:

Step-by-step explanation:

5

The average yield per plant is 4.00 pints, and the standard deviation is approximately 0.35 pints.

To calculate the average yield per plant:

1. Add up all the yields: 4 + 3.5 + 4.5 + 4.2 + 3.8 = 19.0 pints.

2. Divide the total yield by the number of plants (5): 19.0 / 5 = 3.8 pints per plant.

To calculate the standard deviation:

1. Find the mean (average) yield per plant, which we already calculated as 3.8 pints.

2. Calculate the squared differences from the mean for each yield:

  - Plant 1: [tex](4 - 3.8)^2[/tex] = 0.04

  - Plant 2:[tex](3.5 - 3.8)^2[/tex] = 0.09

  - Plant 3: [tex](4.5 - 3.8)^2[/tex] = 0.49

  - Plant 4:[tex](4.2 - 3.8)^2[/tex] = 0.16

  - Plant 5:[tex](3.8 - 3.8)^2[/tex] = 0

3. Calculate the average of these squared differences: (0.04 + 0.09 + 0.49 + 0.16 + 0) / 5 = 0.156.

4. Take the square root of the variance to get the standard deviation: √0.156 ≈ 0.35 pints.

The average yield per plant provides a measure of the central tendency of the data set, indicating the typical yield for one plant.

The standard deviation measures the spread or variability of the data points around the mean. A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation indicates more variability. In this case, the standard deviation of approximately 0.35 pints suggests that the yields are relatively close to the average of 3.8 pints per plant, with minimal variability.

Complete question

A homeowner has 5 cherry tomato plants in her garden. Over the course of the season the yields (in pints of tomatoes per plant) are: Plant 1 2 3 4 5 Yield 4 3.5 4.5 4.2 3.8 What is the average yield per plant, and what is the standard deviation (to two decimal places)?

Answer:

jump discontinuity at x = 0; point discontinuities at x = –2 and x = 8

Step-by-step explanation:

From the graph we can see that there is a whole in the graph at x=-2.

This is referred to as a point discontinuity.

Similarly, there is point discontinuity at x=8.

We can see that both one sided limits at these points are equal but the function is not defined at these points.

At x=0, there is a jump discontinuity. Both one-sided limits exist but are not equal.

The accurate discontinuities for the graph that we have here is jump discontinuity at x = 0; point discontinuities at x = –2 and x = 8.

This is the point in the graph where the values that are supposed to be in the equation would have to jump instead of being continuous. This is the point where there are broken lines.

The points in the graph that we have here can be gotten in the second option of the question.

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Rewriting [tex]z[/tex] in polar form makes this trivial.

[tex]z=|z|e^{i\mathrm{arg}(z)}[/tex]

We have

[tex]|z|=\sqrt{(-1)^2+(-\sqrt3)^2}=2[/tex]

[tex]\mathrm{arg}(z)=\tan^{-1}(-1,-\sqrt3)=-\dfrac{2\pi}3[/tex]

(not to be confused with the standard inverse tangent function [tex]\tan^{-1}x[/tex]. Here [tex]\tan^{-1}(x,y)[/tex] is the inverse tangent function that takes into account position in the coordinate plane; look up "atan2" for more information)

So we have

[tex]z=-1-\sqrt3\,i=2e^{-2\pi/3\,i}[/tex]

Then

[tex]z^6=2^6\left(e^{-2\pi/3\,i}\right)^6=64e^{-4\pi\,i}=64[/tex]

so that [tex]a=64[/tex] and [tex]b=0[/tex].

Answer:

Step-by-step explanation:

Vertically stretched. The action of vertically stretched is accomplished by altering a in

y = a* abs(x)

What that means is that you make a > 1. In this case, a = 2

So far, what you have is

y = 2*abs(x)

Six units down. The action of 6 units down is accomplished by a number added or subtracted to/from absolute(x). down is minus, up is plus.

y = 2*abs(x) - b. Since we are moving down, b<0

y = 2*abs(x) - 6

Four Units Right. This is the tough one because it is anti intuitive. You would think you should be adding something somewhere to get a right hand movement.

Not true.

To move right you subtract something in the brackets.

y = 2*abs(x - 4) - 6

Graph

Just to make things complete, I have graphed this for you. Desmos is wonderful for this kind of problem.

red: y = abs(x)

blue: y = 2*abs(x - 4) - 6

Answer:

Step-by. step explanation:

Answer:

10976π/3 cm³

Step-by-step explanation:

Volume of a sphere is:

V = 4/3 π r³

where r is the radius (half the diameter).

The diameter is 28 cm, so the radius is 14 cm.

V = 4/3 π (14 cm)³

V = 10976π/3 cm³

Final answer:

The volume of a basketball with a diameter of 28 cm is 4,304π cm³ when expressed in terms of π.

Explanation:

To find the volume of a basketball in terms of π (pi), we start by noting that the formula for the volume of a sphere is V = ⅔πr³, where r is the radius of the sphere. Given that the diameter of the basketball is 28 cm, we will first find the radius by dividing the diameter by 2, which gives us a radius of 14 cm. We then plug this into the formula:

V = ⅔π(14 cm)³

After simplifying, we get:

V = ⅔π(2744 cm³)

And finally, the volume V in terms of π is:

V = 4,304π cm³

Answer:

D. 158

Step-by-step explanation:

Fill in the x's with a 15:

[tex]y=-2(15)^2+40(15)+8[/tex]

so y = 158

The best prediction for the number of new cases in year 15 is 158.

Any equation containing one term in which the unknown is squared and no term in which it is raised to a higher power solve for x in the quadratic equation.

Here, Given quadratic equation

                 y = -2x² + 40x + 8

 at put the value of x = 15, we get

               y(15) = -2 X (15)² + 40 X 15 + 8

               y(15) = -2 X 225 + 600 + 8

               y(15) = -450 + 600 + 8

              y(15) = 150 + 8

              y(15) = 158

Thus, the best prediction for the number of new cases in year 15 is 158.

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Answer:

3 3/4

Step-by-step explanation:

I've no idea if this is right, but this is how I'd do it:

For 2 batches you need 2 1/2 cups of blueberry

So, half of that is 1 and 1/4

So the total is 3 3/4

Answer:

The answer is 3 3/4

Step-by-step explanation:

I got it right on my test.

Answer:

f(x) = (x + 1)² + 2 in vertex form ⇒ 3rd answer

Step-by-step explanation:

* Lets revise how to find the vertex form the standard form

- Standard form ⇒ x² + bx + c, where a , b , c are constant

- Vertex form ⇒(x - h)² + k, where h , k are constant and  (h , k) is the

  vertex point (minimum or maximum)  of the function

- At first we must find h and k

- By equating the two forms we can find the value of h and k

* Lets solve the problem

∵ f(x) = x² + 2x + 3 ⇒ standard form

∵ f(x) = (x - h)² + k ⇒ vertex form

- Put them equal each other

∴ x² + 2x + 3 = (x - h)² + k ⇒ open the bracket power 2

∴ x² + 2x + 3 = x² - 2hx + h² + k

- Now compare the like terms in both sides

∵ 2x = -2hx ⇒ cancel x from both sides

∴ 2 = -2h ⇒ divide both sides by -2

∴ -1 = h

∴ The value of h is -1

∵ 3 = h² + k

- Substitute the value of h

∴ 3 = (-1)² + k

∴ 3 = 1 + k ⇒ subtract 1 from both sides

∴ 2 = k

∴ The value of k = 2

- Lets substitute the value of h and k in the vertex form

∴ f(x) = (x - -1)² + 2

∴ f(x) = (x + 1)² + 2

* f(x) = (x + 1)² + 2 in vertex form

You are given h(t) = 64–16t^2.

You need to find h(1.25).

Set up:

h(1.25) = 64 - 16(1.25)^2

Take it from here.

Answer:

Step-by-step explanation:

Given the height of a beach ball rolling off a cliff as a function of time

h = 64 — 16t²

We want to find the average change in height when t = 1.25s

The rate of change of height is the differentiation of the height I.e

∆h/∆t → dh/dt

Therefore

h = 64 — 16t².

Therefore

Differential of a constant is zero

dh/dt = —32t

At t = 1.25s

dh/dt = —32(1.25)

dh/dt = —40 ft/s

So, the height is reducing at a rate of 40ft in every seconds.

Answer:

they both have the equilateral lines on each side

Step-by-step explanation:

Answer:

1/36 or 0.0277... or approximately a 3% chance of rolling a sum greater than 11 (i.e., a sum of 12)

Step-by-step explanation:

Create a 6 x 6 grid on a piece of graph paper. Number the 6 columns 1-6 at the top (for the value of the first die) and number the rows 1-6 on the side (for the value of the second die). You must imagine that you are rolling one die then the other (rather than how people "normally" roll both simultaneously). In the 36 empty boxes in your grid below the column numbers and to the right of the row numbers, put the sums of rolling the die at the top of the column + its corresponding die from the row (for example, use your fingers to match column number 4, say, with row 4 to get a sum of 8). When you've filled out the grid, you will see that a sum of 2 and sum of 12 have the same probability (there's only one way to get a sum of 2 or sum of 12, either 1 + 1 or 6 + 6). But there are many more ways, for example, to get a sum of 7

Answer:

Step-by-step explanation:

A parameter describes a population, and a statistic describes a sample.

The first one is a statistic.  A sample was surveyed.

The second one is a parameter.  It describes the entire soccer team.

The third one is a statistic.  A sample was surveyed.

The fourth one is a parameter.  It describes the entire golf team.

Answer:

Q13. y = sin(2x – π/2); y = - 2cos2x  

Q14. y = 2sin2x -1; y = -2cos(2x – π/2) -1

Step-by-step explanation:

Question 13

(A) Sine function

y = a sin[b(x - h)] + k

y = a sin(bx - bh) + k; bh = phase shift

(1) Amp = 1; a = 1

(2) The graph is symmetrical about the x-axis. k = 0.

(3) Per = π. b = 2

(4) Phase shift = π/2.  

2h =π/2

h = π/4

The equation is

y = sin[2(x – π/4)} or

y = sin(2x – π/2)

B. Cosine function

y = a cos[b(x - h)] + k

y = a cos(bx - bh) + k; bh = phase shift

(1) Amp = 1; a = 1

(2) The graph is symmetrical about the x-axis. k = 0.

(3) Per = π. b = 2

(4) Reflected across x-axis, y ⟶ -y

The equation is y = - 2cos2x  

Question 14

(A) Sine function

(1) Amp = 2; a = 2

(2) Shifted down 1; k = -1

(3) Per = π; b = 2

(4) Phase shift = 0; h = 0

The equation is y = 2sin2x -1

(B) Cosine function

a = 2, b = -1; b = 2

Phase shift = π/2; h = π/4

The equation is

y = -2cos[2(x – π/4)] – 1 or

y = -2cos(2x – π/2) - 1

2/3

There are 4 green marbles and 6 red marbles, making 10 green or red marbles.  The probability of drawing a red or a green marble is then 10/15 total marbles, which can be simplified to 2/3 by dividing the numerator and denominator each by 5.

Answer:

b. [tex]\frac{p-1700}{10}[/tex]

Step-by-step explanation:

We have that the server's salary can be expressed as

[tex]p = $1700+$10t[/tex]

Counting the $ 1700 dollars as the fixed amount of your salary and contemplating the variable salary that would be $ 10 * t being t the number of overtime worked  

from this expression we can clear the number of overtime worked by clearing t, so I finally have

[tex]p=$1700+$10t\\ $10t = p - $1700 \\ t= \frac{p-1700}{10}[/tex]

Done

For this case we have that by definition, the circumference of a circle is given by:

[tex]C = \pi * d[/tex]

Where:

d: It is the diameter of the circle

They tell us that:

[tex]d = \frac {35} {2}\\\pi = \frac {22} {7}[/tex]

Substituting:

[tex]C = \frac {22} {7} * \frac {35} {2}\\C = \frac {770} {14}\\C = 55 \ cm[/tex]

Thus, the circumference of the circle is 55 centimeters.

Answer:

Option B

Answer:

55 im not sure

Step-by-step explanation:

Answer:

The expression for 8y - 2(y+4) is 6y - 8

Step-by-step explanation:

An expression is a mathematical phrase which can contain symbols like a,b,c,x,y,z or numbers like 1,2,3,4,5, etc, or both. Expression is one sided where as equation in mathematics is a two sided expression. There is no left or right side in an Expression, there is always one side.

The given phrase is simplified as follows:

8y - 2(y+4)

8y -2y - 8

6y - 8

Your answer is A) Acute

Answer:

Read the explanation

Step-by-step explanation:

- The Fibonacci Sequence is the series of numbers:

 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

- The next number is found by adding up the two numbers before it

# Example:

- The 2 is found by adding the two numbers before it (1+1)

- The 3 is found by adding the two numbers before it (1+2),

- The 5 is (2+3) and so on

- The rule of it is x(n) = x(n-1) + x(n-2) where

# x(n) is term number ⇒ n

# x(n-1) is the previous term ⇒ (n-1)

# x(n-2) is the term before that ⇒ (n-2)

- Example: the 8th term is  the 7th term plus the 6th term:

 x(8) = x(7) + x(6)

# Note: When we take any two consecutive Fibonacci Numbers,

 their ratio is very close to the Golden Ratio φ

- The Golden Ratio φ is approximately 1.618034...

- We can calculate any Fibonacci Number using the Golden Ratio:

  x(n) = [φ^n - (1 - φ)^n]/√5

- The answer is a whole number, exactly equal to the addition of the

  previous two terms.

# There is an interesting patterns in Fibonacci sequence:

  every nth number is a multiple of x(n)

- Example:

* x3 = 2 ⇒ every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, ...)

* x4 = 3 ⇒ Every 4th number is a multiple of 3 (3, 21, 144, ...)

* x5 = 5 ⇒ Every 5th number is a multiple of 5 (5, 55, 610, ...)

- The man who invented it.

 His real name was Leonardo Pisano Bogollo, and he lived between

 1170 and 1250 in Italy.

- Fibonacci was his nickname

- Fibonacci sequence is it used for:

# Reflects patterns of growth spirals (a spiral curve , shape , pattern ,

  object) found in nature

# It is the closest approximation in integers to the logarithmic spiral

  series, which follows the same rule as the Fibonacci sequence

The Fibonacci sequence is a series of numbers important in mathematics and nature, introduced to European mathematics by Leonardo of Pisa, also known as Fibonacci. It describes various natural phenomena, appears in plants' growth patterns, contributes to arts and sciences, and has practical uses in algorithms and market analysis.

The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, and so forth. This mathematical concept was brought to European mathematics by Leonardo of Pisa, known as Fibonacci. He described this sequence in his book 'Liber Abaci' in 1202, although the sequence had been previously described in Indian mathematics.

The Fibonacci sequence is not just a number theory curiosity but has many applications and appears in the natural world. Notably, it is used to describe various phenomena in quantum computing, stock market analysis, and in nature, such as the branching of trees, the arrangement of leaves on a stem, or the fruitlets of a pineapple. In the context of spirals and botany, the mechanism of the spirals of plants follows the Fibonacci sequence, as explored in resources like 'Doodling in Math: Spirals, Fibonacci, and Being a Plant' by Vihart and 'How to Count the Spirals' by MoMath: National Museum of Mathematics.

The Fibonacci sequence also finds itself being applied to areas such as computer algorithms, especially in recursive algorithms where an operation needs to be repeated on a smaller scale. It also helps build a bridge between mathematics and the arts, showing how mathematics can inspire beautiful patterns and structures in man-made and natural art forms. The prevalence of the Fibonacci sequence and the golden ratio in aesthetic compositions demonstrates the union between form and number that fascinates both mathematicians and artists alike.

Answer:

D. (–3, –6) and (5, 10)

Step-by-step explanation:

The system has equations:

[tex]y=2x[/tex]...(1)

and

[tex]y=x^2-15[/tex]...(2)

Equate both equations;

[tex]x^2-15=2x[/tex]

Rewrite in standard form:

[tex]x^2-2x-15=0[/tex]

We factor to obtain:

[tex](x+3)(x-5)=0[/tex]

By the zero product principle;

[tex](x+3)=0,(x-5)=0[/tex]

[tex]\implies x=-3,x=5[/tex]

When x=-3, y=2(-3)=-6

This yields the ordered pair (-3,-6).

When x=5, y=2(5)=10

This yields the ordered pair (5,10).

The correct choice is D.

Answer:

B. Multiply 5 times 3. Multiply 5 times 4. Add the two products

Step-by-step explanation:

Right from the start the question is only asking for squash plants so the other two rows don't need to be counted.

7-2 = 5

In each row there's 3 yellow squash = 5*3 = 15

And 4 green squash = 5*4 = 20

which means there's 35 squash total in the garden.