Reuben L. Goldberg and His Machines

The Artist and His Work
Reuben (Rube) L. Goldberg was an American cartoonist. He is best known for drawing elaborate machines. His drawings showed machines with wheels, gears, handles, pails, and other interesting devices. The machines in Goldberg's drawings were designed to take a simple task and make it complex. The contraptions he drew 80 years ago were so funny that they continue to inspire young scientists and engineers today.

Goldberg's Career
Goldberg's cartoons appeared in several respected New York City newspapers, including the New York Evening Journal and the New York Evening Mail. The cartoonist also wrote a feature film called Soup to Nuts, which was the first film starring the Three Stooges. Goldberg's machines appear in the film.

How to Build a Cause-Effect Machine
Every year students build machines based on Goldberg's designs for school projects, for competitions, or simply for fun. Goldberg's machines operate based on cause and effect. Students and engineers of all ages can learn to build these elaborate cause-effect machines. The key to building one of these machines is to understand how to use gravity and force to push and pull objects along a path. Reading up on these concepts is a good place to start. After that, building the machines is all about having fun. Find a simple task for the machine to do. Then, find a complicated way for the machine to do it.

The Cause-Effect Machine in Popular Culture
Cause-effect machines are incredibly fun to watch. For this reason, they have found their way into Hollywood movies such as Back to the Future and The Goonies. The machines also inspired a board game for kids called Mouse Trap. Clearly, Goldberg's machines have a place in our popular culture.
4
Which section develops the idea that Goldberg's machines help inspire young scientists and engineers?
A.
How to Build a Cause-Effect Machine
B.
The Artist and His Work
C.
Goldberg's Career
D.
The Cause-Effect Machine in Popular Culture

The mean of the z scores after converting the pulse rates to z scores is 0, and the standard deviation of the z scores is 1. The units of the corresponding z scores are dimensionless.

The process of converting the pulse rates to z scores involves subtracting the mean and dividing by the standard deviation. This transformation standardizes the data and makes it comparable across different distributions. In the case of z scores, the mean is always 0, and the standard deviation is always 1. Therefore, the mean and standard deviation of the z scores in this scenario are 0 and 1, respectively.

Since z scores are calculated by subtracting the mean and dividing by the standard deviation, the resulting values are dimensionless. They represent the number of standard deviations away from the mean. Therefore, the units of the corresponding z scores are neither "beats per minute," "beats," nor "minutes per beat." They are simply numerical values without specific units.

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The correct one is Eden, the exponents should be added.

The mistakes are that Erin multiplies the exponents and Emma multiplies the bases.

Remember that when we have the product of two powers with the same base, we just need to add the exponents, so:

\(x^n*x^m = x^{n + m}\)

With that in mind, we can see that the one who did the operation correctly is Eden, because:

\(6^2*6^5 = 6^{2 + 5} = 6^7\)

The mistake for Erin is that she multiplied the exponents, while the problem for Emma is that she also multiplied the bases.

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The four points form the vertices of a quadrilateral is w° (1, 0), w (1, √3), w² (-2, √3), w' (1, -√3)

Let's analyze the complex number w and plot its powers and conjugate on an Argand diagram.

Given w = 2(cos(π/3) + i sin(π/3)), we can find w°, w², and w'.

1. w° is the 0th power of w, which is always 1 (1 + 0i) for any non-zero complex number.

2. w² can be found using De Moivre's theorem:
w² = 2²(cos(2π/3) + i sin(2π/3)) = 4(-1/2 + i√3/2).

3. w' is the complex conjugate of w:
w' = 2(cos(π/3) - i sin(π/3)) = 2(1/2 - i√3/2).

Now, let's plot these points on the Argand diagram:
- w° (1, 0)
- w (1, √3)
- w² (-2, √3)
- w' (1, -√3)

These four points form the vertices of a quadrilateral.

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The probability that a valve was ground twice can be calculated using Bayes' theorem. Let G1 be the event that a valve is reground once and G2 be the event that a valve is reground twice. Then, the probability of a valve being scrapped given that it was reground once is 19%, and the probability of a valve meeting the specification given that it was reground twice is 100%. Therefore, using Bayes' theorem, we can calculate the probability of a valve being ground twice given that it was scrapped as (0.14 x 0.19) / ((0.14 x 0.19) + (0.24 x 0.81)) = 0.029. Thus, the probability that a valve was ground twice given that it was scrapped is 2.9%.

Bayes' theorem is a mathematical formula used to calculate conditional probabilities. It states that the probability of an event A given an event B is equal to the probability of event B given event A, multiplied by the probability of event A, divided by the probability of event B. In this problem, we want to calculate the probability of a valve being ground twice given that it was scrapped.
To apply Bayes' theorem, we first need to identify the relevant probabilities. We are given that after the first grind, 62% of the valves meet the specification, 24% are reground once, and 14% are scrapped. We are also given that of those valves that are reground, 81% meet the specification, and 19% are scrapped.
Let G1 be the event that a valve is reground once and G2 be the event that a valve is reground twice. Then, the probability of a valve being scrapped given that it was reground once is 19%. The probability of a valve meeting the specification given that it was reground twice is 100%, since all valves that are reground twice are guaranteed to meet the specification.
Using Bayes' theorem, we can calculate the probability of a valve being ground twice given that it was scrapped as follows:
P(G2|scrapped) = P(scrapped|G2) x P(G2) / P(scrapped)
where P(scrapped|G2) is the probability of a valve being scrapped given that it was reground twice, P(G2) is the probability of a valve being reground twice, and P(scrapped) is the probability of a valve being scrapped.
We already know that P(scrapped|G1) = 0.19, P(scrapped|G2) = 0, P(G1) = 0.24, P(G2) = (1 - 0.62 - 0.24 - 0.14) x P(G1) = 0.027, and P(scrapped) = 0.14.
Plugging in the values, we get:
P(G2|scrapped) = (0 x 0.027) / ((0.14 x 0.19) + (0.24 x 0.81)) = 0.029
Thus, the probability that a valve was ground twice given that it was scrapped is 2.9%.

In summary, we can use Bayes' theorem to calculate the probability of a valve being ground twice given that it was scrapped. We first identify the relevant probabilities, such as the probability of a valve being scrapped given that it was reground once or twice. We then apply Bayes' theorem to obtain the desired probability. In this case, the probability that a valve was ground twice given that it was scrapped is 2.9%.

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First make the graph of the functions, then  if the two graphs are symmetric with respect to the line y = x (mirror images over y = x ), then they are inverse functions.

Symmetric

A figure or shape that can be divided into two equal parts by a line is called symmetric figures.

Inverse Functions

The inverse function of a function f  is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by F^-1.

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Answer:

8x + 2y = 10  ==>  y = -4x + 5

12x - 3y = -25​  ==>  y = 4x + 25/3

Step-by-step explanation:

8x + 2y = 10  ==> solve for y

2y = 10 - 8x  ==> subtract 8x on both sides

(2y = 10 - 8x) / 2 ==> divide the equation by 2 to make 2y into y

y = 5 - 4x  ==> simplify

y = -4x + 5  ==> put the equation in y=mx+b form

12x - 3y = -25​  ==> solve for y

12x = -25 + 3y  ==> make y positive by adding 3y on both sides

12x + 25 = 3y ==> add 25 on both sides to isolate y

(3y = 12x + 25) / 3  ==> divide the equation by 3 to make 3y into y

y = 4x + 25/3 ==> simplify

Answer:

a) x= 82°

Step-by-step explanation:

Please see the attached image for the full solution.

Answer:

The value of x is 82⁰

Step-by-step explanation:

As we know that,

The sum of two opposite interior angle of triangle is equal to it's exterior angle :

Then,

\( \sf \longrightarrow{x + 21 \degree = 103 \degree}\)

\( \sf \longrightarrow{x = 103 \degree - 21 \degree}\)

\( \sf \longrightarrow{x = 82 \degree}\)

Hence, the value of x is 82⁰.

The x-intercepts of the parabola are (3,0) and (-2,0)

Knowing the roots of the quadratic formula you can express the factorized form:

Next, solve the multiplication of the parentheses terms by multiplying each term of the first parentheses by each term of the second parentheses

The probability of not selecting a green marble is equal to the total number of non-green marbles in the bag divided by the total number of marbles in the bag.

To calculate the probability: there are three green marbles out of a total of twenty-five marbles, so the probability of selecting a green marble is 3/25.

The likelihood of not selecting a green marble is then 1 - 3/25 = 22/25.

This is equal to 22/25 * 100 = 88% as a percentage.

As a result, P(not green) = 88%

Probability denotes the possibility of something happening. It is a mathematical branch that deals with the onset of a random event. The value ranges from zero to one. Probability has been tried to introduce in mathematics to predict the probability of events occurring. Probability is defined as the degree to which something is likely to occur. This is the fundamental probability theory, which is also used in probability distribution, in which you will learn about the possible outcomes of a random experiment.

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Answer:

179 sq feet

Step-by-step explanation:

What is the approximate area of the walkway?

area of walkway = area of walkway +  fountain - area of walkway

Area of a circle = πr²

Where : = π = pi = 22/7

R = radius  

radius of the fountain = 16/2 = 8

area of walkway +  fountain = (22/7) x (8 + 3)^2 = 380.3

area of fountain = 22/7 x 8^2 = 201.1

area of walkway = 179

Given:

In an asteroid field, 75% of the asteroids are carbonaceous asteroids.

There are 375,000 carbonaceous asteroids in the asteroid field.

To find:

The number of asteroids that are not carbonaceous.

Solution:

Let the total number of asteroids be x.

According to question,

\(75\%\text{ of }x = 375000\)

\(\dfrac{75}{100}x=375000\)

\(0.75x=375000\)

Divide both sides by 0.75.

\(x=\dfrac{375000}{0.75}\)

\(x=500000\)

Asteroids not carbonaceous = Total asteroids - Carbonaceous asteroids

                                                \(=500000-375000\)

                                                \(=125000\)

Therefore, 125,000 asteroids are not carbonaceous.

Answer:

Step-by-step explanation:

Vertices of the given triangle ABC are A(-1, 4), B(-1, 1) and C(-3, 1)

Equation of the mirror line → y = x

Rule to reflect these vertices over the mirror line y = x,

(x, y) → (y, x)

Therefore, image points of the vertices will be,

A(-1, 4) → A'(4, -1)

B(-1, 1) → B'(1, -1)

C(-3, 1) → C'(1, -3)

Now we can plot these image points to get the image triangle A'B'C'.

Hello!

\(\large\boxed{2x + y = 3}\)

Begin by finding the slope that would be perpendicular to the given equation.

A perpendicular line has a slope that is the opposite reciprocal of the original. Therefore:

1/2 --> -2

Find the answer choice that contains a slope of -2 when rewritten in the form y = mx + b:

Choice 1:

x + 2y = 2

Move x to the other side:

2y = -x + 2

Divide both sides by 2:

y = -1/2x + 1. This choice is incorrect.

Choice 2:

x - 2y = -12

Move x to the other side:

-2y = -x - 12

Divide both sides by -2:

y = 1/2x + 6. This choice is incorrect.

Choice 3:

2x + y = 3

Move 2x to the other side:

y = -2x + 3. This contains a slope of -2, which is correct.

Choice 4:

y - 2x = -5

Move -2x to the opposite side:

y = 2x - 5. This contains a slope of positive 2, so it is incorrect.

The correct answer is 2x + y = 3.

Answer:

D. y-2x=-5

Step-by-step explanation:

Answer:

The Great Western Trail = 4,455 miles

Step-by-step explanation:

The Iditarod Trail = 1,025 miles long

The Great Western Trail is 355 miles longer than 4 times the length of the Iditarod Trail.

The Great Western Trail = (4 × 1,025) miles + 355 miles

= 4,100 miles + 355 miles

= 4,455 miles

The Great Western Trail = 4,455 miles

9514 1404 393

Answer:

  C

Step-by-step explanation:

All of the offered answer choices are dilations. When a figure is dilated about the origin, every individual coordinate is multiplied by the dilation factor.

So, the dilation factor can be found by dividing an image coordinate by the pre-image coordinate.

  I'x/Ix = -3/-2 = 1.5

The dilation is 1.5 about the origin.  . . .  matches C

Answer:

+9 and -9

Step-by-step explanation:

Using the Factor Theorem, the polynomial is factored as follows:

f(x) = (x - 2)(x + 1)(x² + 2).

The Factor Theorem states that a polynomial function with roots(also called zeros) \(x_1, x_2, \codts, x_n\) is given by the following rule.

\(f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)\)

In which a is the leading coefficient of the polynomial, determining if it is positive(a positive) or negative(a negative).

In this problem, the polynomial is given by:

f(x) = x^4 - x³ - 2x - 4.

This polynomial is factored applying the inverse factor theorem, finding the roots of the polynomial, which are given as follows:

For the complex roots, we have that 1.41² = 2, hence the term corresponding to the two complex roots is:

x² + 2.

Then the complete factored polynomial is:

f(x) = (x - 2)(x + 1)(x² + 2).

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In the right triangle figure in the image the value of the missing sides are

The sides of the right triangle is worked using SOH CAH TOA

Sin = opposite / hypotenuse - SOH

Cos = adjacent / hypotenuse - CAH

Tan = opposite / adjacent - TOA

For side d we use tan

tan 30 = d/7

d = 7 * tan 30

d = 4.04

For side b using cos

cos 45 = 2/b

b = 2 / (cos 45)

b = 2.828

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Answer:

7/5

Step-by-step explanation:

What is 1.4 in fraction form? 7/5 is the fraction form of 1.4. 1.4 as a Fraction in simplest form to convert 1.4 to a fraction and simplify to the lowest form.

------------

14/10 = 1.4

7 is half of 14

5 is half of 10

The simplest form of given number is  \(\frac{7}{5} \).

Given number is, 1.4

Converted into fraction,  \(=\frac{14}{10} \)

It is observed that, 2 is common in both numerator and denominator.

               \(\frac{14}{10} =\frac{2*7}{2*5}=\frac{7}{5} \)

Hence, The simplest form of given number is  \(\frac{7}{5} \).

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Answer:

k = - 8

Step-by-step explanation:

given that (x - a) is a factor of f(x) , then f(a) = 0

given

(x - 1) is a factor of f(x) then f(1) = 0 , that is

3(1)³ + 5(1) + k = 0

3(1) + 5 + k = 0

3 + 5 + k = 0

8 + k = 0 ( subtract 8 from both sides )

k = - 8

As per similar rule:

So:

We need to find the area. So we multiply the dimensions.

3x1/2=1.5

1.5x4 1/2=6.75

Decimal was my go-to so we need to convert back to a fraction.

6.75= 6 3/4

So the answer is B.

---

hope it helps

Answer:

\(5)8 \sqrt[6]{?} \)

allá paosgesusussj siisveveu días dbd dusv

The vertices of the dilated polygon A'B'C'D' are A'(-3, 4.5), B'(-1.5, 1.5), C'(3, -1.5), and D'(3, 3). the correct option is (A) A′(−3, 4.5), B′(−1.5, 1.5), C′(3, −1.5), D′(3, 3).

What is polygon?

A polygon is a closed plane figure that is formed by three or more straight sides, which are connected end-to-end to create a closed shape.

To find the vertices of the dilated polygon, we need to apply the scale factor of 3/4 to each of the original vertices. This will result in a new set of coordinates for each vertex.

The coordinates of point A' can be found by multiplying the x-coordinate and y-coordinate of A by 3/4:

x-coordinate of A' = -4 × 3/4 = -3

y-coordinate of A' = 6 × 3/4 = 4.5

Therefore, A' has coordinates (-3, 4.5).

Using the same method, we can find the coordinates of the other vertices:

B':

x-coordinate of B' = -2 × 3/4 = -1.5

y-coordinate of B' = 2 × 3/4 = 1.5

Therefore, B' has coordinates (-1.5, 1.5).

C':

x-coordinate of C' = 4 × 3/4 = 3

y-coordinate of C' = -2 × 3/4 = -1.5

Therefore, C' has coordinates (3, -1.5).

D':

x-coordinate of D' = 4 × 3/4 = 3

y-coordinate of D' = 4 × 3/4 = 3

Therefore, D' has coordinates (3, 3).

Thus, the vertices of the dilated polygon A'B'C'D' are A'(-3, 4.5), B'(-1.5, 1.5), C'(3, -1.5), and D'(3, 3). Therefore, the correct option is (A) A′(−3, 4.5), B′(−1.5, 1.5), C′(3, −1.5), D′(3, 3).

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(a) To plot the log-likelihood function of the parametric model, we can use the expression derived earlier:log(L(θ, α)) = 4 * log(θ) + 4 * log(1 - θ - α) + 2 * log(α) (b) The maximum likelihood estimate of θ is 0.4 and the maximum likelihood estimate of α is 0.2. (c) The empirical pmf for the data is P(Garry wins) = 0.4, P(Draw) = 0.4, and P(Anish wins) = 0.2.

To solve this problem, let's define the likelihood function based on the given information. We have 10 games in total, and Garry wins 4, they draw 4, and Anish wins 2. Let's assume that the games are independent and in each game, Garry has a probability θ of winning, and Anish has a probability α of winning.

(a) The likelihood function L(θ, α) can be calculated as the product of the probabilities of each outcome:

L(θ, α) = P(Garry wins)⁴ * P(Draw)⁴* P(Anish wins)²

Since the games are independent, we can calculate the probability of each outcome using the probabilities θ and α:

P(Garry wins) = θ

P(Draw) = (1 - θ - α)

P(Anish wins) = α

Therefore, the likelihood function becomes:

L(θ, α) = θ⁴* (1 - θ - α)⁴ * α²

To plot the log-likelihood function, we take the logarithm of the likelihood function:

log(L(θ, α)) = 4 * log(θ) + 4 * log(1 - θ - α) + 2 * log(α)

(b) To find the maximum likelihood estimate of θ and α, we need to maximize the log-likelihood function. We can do this by taking partial derivatives with respect to θ and α and setting them equal to zero.

Differentiating log(L(θ, α)) with respect to θ:

d/dθ [log(L(θ, α))] = 4/θ - 4/(1 - θ - α)

Setting it equal to zero:

4/θ - 4/(1 - θ - α) = 0

Simplifying:

1/θ = 1/(1 - θ - α)

θ = 1 - θ - α

2θ + α = 1

Similarly, differentiating log(L(θ, α)) with respect to α:

d/dα [log(L(θ, α))] = -4/(1 - θ - α) + 2/α

Setting it equal to zero:

-4/(1 - θ - α) + 2/α = 0

Simplifying:

-2/(1 - θ - α) + 1/α = 0

2/α = 1/(1 - θ - α)

α = 2(1 - θ - α)

α = 2 - 2θ - 2α

2θ + 3α = 2

Solving these two equations:

2θ + α = 1

2θ + 3α = 2

By solving these equations, we can find the maximum likelihood estimates of θ and α.

(c) To model the data as realizations from a discrete random variable, we can calculate the empirical probability mass function (pmf). The empirical pmf is obtained by counting the frequencies of each outcome and dividing by the total number of games.

Based on the given information, Garry wins 4 games, they draw 4 games, and Anish wins 2 games out of a total of 10 games. The empirical pmf is as follows:

P(Garry wins) = 4/10 = 0.4

P(Draw) = 4/10 = 0.4

P(Anish wins) = 2/10 = 0.2

Comparing the nonparametric model (empirical pmf) to the parametric model (likelihood function) allows us to assess the fit of the parametric model to the observed data.

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Answer:

a: 2.234

Step-by-step explanation:

i checked after i took the test. its right, enjoy

The size of the square cut from each corner will be 1.369 inches. The correct option is C.

Geometry is one of the oldest branches of mathematics, along with arithmetic. It is concerned with spatial properties such as figure distance, shape, size, and relative position.

Given that a box with no top that has a volume of 1000 cubic inches is to be constructed from a 22 x 30-inch sheet of cardboard by cutting squares of equal size from each corner and folding up the flaps.

The size of the square cut from each corner will be calculated as,

Volume = L x H x W

1000 = 22 x 30 x H

H = ( 1000 ) / ( 660 )

H = 1.538 or 1.369 inches

Therefore, the size will be 1.369 inches.

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Answer:

D  (-6, -5)

Step-by-step explanation:

We need to create a linear function for the set of coordinates, input the x-values of the possible options into the function, then see which one gives the correct y-value.

Pick 2 coordinate pairs from the set:

Let \((x_1,y_1)\) = (0, 4)

Let \((x_2,y_2)\) = (-2, 1)

Use slope formula: \(m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{1-4}{-2-0}=\dfrac32\)

Use equation of a line in point-slope form to create the linear function:

      \(y-y_1=m(x-x_1)\)

\(\implies y-4=\dfrac32(x-0)\)

\(\implies y=\dfrac32x+4\)

Therefore, the linear function is \(f(x)=\dfrac32x+4\)

Inputting the x-values of the possible options, (-6, -5) is the only coordinate pair that is correct:

\(f(-6)=\dfrac32(-6)+4=-5\)

Slope:-

\(\\ \tt\hookrightarrow m=\dfrac{1-4}{-2}=\dfrac{-3}{-2}=\dfrac{3}{2}\)

Equation of line in point slope form

\(\\ \tt\hookrightarrow y-4=3/2(x)\)

\(\\ \tt\hookrightarrow y=3/2x+4\)

Check out option D

\(\\ \tt\hookrightarrow -5=3/2(-6)+4\)

\(\\ \tt\hookrightarrow -5=-9+4\)

\(\\ \tt\hookrightarrow -5=-5\)

Hence option D is correct