7B Math: Efficiency Expert Portfolio
Directions:
You will mathematically and experimentally show how 4 figures made from the SAME size paper will produce different volumes.
Take 4 sheets of paper (I used 8 ½ x 11 printer paper you will use 8x10 in) and create:
-A: tall rectangular prism
-B: short rectangular prism
-C: tall cylinder
-D: short cylinder

Mathematically you will figure the dimensions of your shapes just using your paper dimensions versus a ruler.


A: B: C: D:






Volume of Prism: l x w x h
Volume of Cylinder: r x r x 3.14 x h
So we will use our paper dimensions to find the shape measurements needed to find the volume of each one.
A and B….the prisms are easy. We just take the dimension and divide it by 4 to find our measurement.
C and D ….the cylinders are a bit more tricky. We know the circumference of our circle tops and the heights of our cylinders but we need to find the radius of our circle top.
Circumference = diameter x 3.14
11 = d x 3.14
Divide both sides by 3.14 to find the diameter
11/3.14 = d
3.5 = d but in finding the volume of the cylinder we need the radius.
d/2 = r 3.5/2 = 1.75 in

Answer:  The mean of data set K is greater than the mean of data set L.

The median of data set K is greater than the median of data set L

Step-by-step explanation:

we need to calculate the mean (average) of each data set and compare them.

Mean of data set K = (28 + 145 + 112 + 57 + 130) / 5 = 112.4

Mean of data set L = (141 + 171 + 68 + 124 + 43) / 5 = 109.4

'Arranging the values in data set K in ascending order, we get:

28, 57, 112, 130, 145

The median of data set K is the middle value, which is 112.

The mean of data set L is 109.4.

Answer:

y=-0.5x+3

Step-by-step explanation:

The equation of a line is

y=mx+b, with m being the slope, and b being the y-intercept

The slope is -0.5, and the y intercept is 3

so:

y=-0.5x+3

Answer:

y= -1/2x+3

Expiation:

you can find the equation between two points by using the slope formula and the slope intercept form which is y=mx+b.

Answer:  $1.80

Step-by-step explanation: 15% of 12 is 1.8

Answer: 16.1 cm

Step-by-step explanation:

Attached below is a diagram of the triangle.

If you have two legs, you can find the hypotenuse of the right triangle by using the Pythagorean Theorem.

Pythagorean Theorem: a² + b² = c²

Let's set

a = 16 cm

b = 2 cm

So we get,

16² + 2² = c²

256 + 4 = c²

260 = c²

Take the square root of both sides to get c. Since you are taking the square root, there are two answers, positive and negative, but a side length can not be negative.

[tex]\sqrt{260}[/tex] = c

Input this into your calculator, and you get

c = 16.1245155 cm

Rounding to the nearest tenth you get c = 16.1 cm

The sets of transformations that result in the same image when performed in any order are:

Translation, dilation with center (0,0)

Two translations

In mathematics, a transformation refers to a process that changes the size, shape, position, or orientation of a geometric figure or an object in a coordinate plane or space. There are several types of transformations, such as translation, rotation, reflection, and dilation, which are used to alter the original figure to create a new one that is similar or congruent to the original. Transformations are used in geometry, algebra, and other branches of mathematics to study and understand various properties of geometric shapes and objects.

Here,

For the first set, if we perform a translation followed by a dilation with center (0,0), the result will be the same as if we perform the dilation first followed by the translation. This is because the dilation does not change the direction of the translation, and the center of dilation is at the origin, so the translation is unaffected.

For the second set, any two translations can be combined to form a single translation, so the order in which they are performed does not matter.

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AC and BD are  the diagonals of the MNOP, the diagonals of the rectangle MNOP bisect each other and are congruent.

 A rectangle and a rectangle  are quadrilaterals that share the following properties: Opposite sides are equal. Two diagonals are the same length. It has two  class 2 reflection symmetries and a rotational symmetry (through 180°).  To show that the diagonals of a rectangle bisect each other and are congruent, we can use the following proof:

Proof:

Let MNOP be a rectangle whose diagonal MO and diagonal NP intersect at  Q.

First we  show that the diagonals bisect each other. Since MNOP is a rectangle, we know that opposite sides are parallel and congruent. Therefore, we can draw segment MP and segment NO which are both parallel  and congruent to each other.

Now consider the triangle MON. Since segment MP is parallel to segment NO, we know that angles MNO and MON are alternate interior angles and are congruent. Similarly, the angles NMO and ONM are also congruent. Therefore, triangle MON is an isosceles triangle with legs MO and NO. This means that the height drawn from Q to the side MN (which is the perpendicular bisector of MO and NO) bisects MO and NO. Similarly, the height drawn from Q to the side OP bisects OP. Therefore we have shown that the diagonals of a rectangle bisect each other.

 Next, we  show that the diagonals are congruent. Since MNOP is a rectangle, we know that opposite sides are parallel and congruent. Therefore, MP is compatible with NO and MO is compatible with NP. Using the fact that the diagonals bisect each other, we can write:

MQ = QO (because the height drawn by Q to the side MN divides MO and NO)

and

NQ = QP (because the height drawn from Q to the side OP divides NP and MO)

Combining these two equations, we get:

MQ + NQ = QO + QP

But we also know that MO is NP-consistent, so we can substitute:

MQ+ NQ = MO + MO

Simplification:  

MQ + NQ = 2MO

Therefore, we have shown that the diagonals of a rectangle are congruent.  Therefore, we have shown that the diagonals of a rectangle bisect each other and are congruent as necessary.

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Answer: 773%

Step-by-step explanation: 7.725666118 x 100 = 772.5666118

772.5666118 = 773 (rounded).

So, 7.725666118 as a percentage is 773%

For example, 50% is written in decimal form as 0.5

And, 0.5 x 100 = 50.

Multiplying by 100 is the same thing as moving the decimal two places to the right.

Answer:Answer: A. tennis=17 , soccer= 12, baseball=27, basketball=44

BAR GRAPH make sure its not a histogram!!

Step-by-step explanation:

just took the test

Answer to these fraction is 31/151

Describe fraction.

Any numerical value that is not a whole number is referred to as a fraction. It symbolizes a portion of something or several portions of something. The numerator and the denominator are the two components of a fraction. The denominator indicates how many equally sized pieces make up a whole, whereas the numerator indicates how many equally sized parts are picked from the whole or collection.

To divide a fraction, all you need to do is flip its numerator and denominator, multiply the outcome by the other fraction, and simplify1.

In light of this, we have:

SUM:

(13/5) + (-12/7) = (13× 7 + -12 ×5) / (5 ×7)

= (91 - 60) / 35 = 31/35

DIFFERENCE:

(13/5) - (-12/7) = (13× 7 - -12× 5) / (5× 7)

= (91 + 60) / 35 = 151/35

Dividing these two fractions:

(31/35) ÷ (151/35) = (31/35)×(35/151) = 31/151

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Answer:

d

Step-by-step explanation:

Answer:

c

Step-by-step explanation:

c

Answer:

10 oranges

Step-by-step explanation:

We Know

A grocery store sells a bag of 6 oranges for $4.80

4.80 / 6 = $0.80 for one orange

If Austin spent $8.00 on oranges, how many did he buy?

We Take

8 / 0.80 = 10 oranges

So, he bought 10 oranges.

The coordinates of B when it is given that A(5,-2) and M(12,6) is (29, 14).

If M is the mid-point of line segment AB, and the coordinates of A = (5,-2) while the coordinates of B = (12,6), then the coordinates of M = (8.5,2), which is what you stated that you got.

You are provided with M = (12,6) so you need to identify the coordinates of B. So, just ask yourself how did I get from A to M ? The answer is move 7 right and 8 up since 5 + 7 = 12 while -2 + 8 = 6. All you have to do is add 7 more to the x-value and 8 more to the y-value to obtain the correct answer of (19,14).

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Find the coordinates of B when it is given that A(5,-2) and M(12,6),

Answer:

Based on the information in the problem, you would have a total of 16 cards, with 6 being face cards, 1 being an ace, and 9 being numbered cards.

Step-by-step explanation:

According to the issue:

The deck contains 52 distinct cards.

There are 36 numbered cards ranging from 2 to 9.

There are four aces in the deck.

There are 12 face cards (including the jack, queen, and king).

If you draw half of the face cards, one ace, and one-fourth of the numbered cards, you will get the following:

Face cards in half: 1/2 * 12 = 6 face cards

1 ace: 1 ace

One-fourth of a deck of numbered cards: 1/4 * 36 = 9 decks of numbered cards

When you add these up, you get a total of 6 + 1 + 9 = 16 cards.

Based on the information in the problem, you would have a total of 16 cards, with 6 being face cards, 1 being an ace, and 9 being numbered cards.

Workout of the fastest person's total time is 2 minutes 34 seconds.

Time is a concept used to describe the sequence and duration of events or changes that occur in the universe. It is a measure of the interval between two events or occurrences, and it allows us to describe and understand the order and duration of events.

In physics, time is often described as a dimension, along with the three spatial dimensions, and is often measured in units such as seconds, minutes, or hours. Time is also relative, meaning that its perception and measurement can vary depending on factors such as motion and gravity.

Overall, time is a fundamental concept in many fields, including physics, philosophy, and everyday life, and its study and understanding are crucial to our understanding of the world around us.

According to the given information:

From the given table we can observe that Ben is the fastest

Total time =53.44 + 49.79+ 50.50

                = 153.73 seconds.

Hence, total time = 2 minutes 34 seconds

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Answer:

The answer for the Area is 8

Step-by-step explanation:

Area of semi circle =1/2pir²

r=d/2=8/2=4

A=1/2×4²

A=1/2×4×4

A=2×4

A=8

Answer:

32 or 100.53

Step-by-step explanation:

*assuming 8 is radius and not diameter*

A = [tex]r^{2}[/tex]

r = 8

[tex]8^{2} = 64[/tex]

A = 64 or ≈201.062

divide by 2 for half of the circle

32 or 100.53

The container can contains 3,000 pounds of chemicals.

One cubic foot of a chemical weighs 50 pounds which means 1 ft³ of chemical = 60 pounds

We must  find how many pounds in the container whose dimension 6 ft, 4 ft and 2.5 ft. So, the l = 6 ft, b = 4 ft and h = 2.5 ft.

Volume of box = L*B*H

Volume of box = 6 x 4 x 2.5

Volume of box = 60 ft³

From 1 ft³ of chemical = 60 pounds; then, the 50 ft³ of chemical will be:

= 50 x 60

= 3,000 pounds

Full question "One cubic foot of a chemical weighs 60 pounds. How many pounds of the chemical can a container with the dimensions shown hold? The base is 6, the width is 4, and the height is 2.5"

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Answer:

Part A:  1/2 or 0.5

Part B: heads is 8, tails is 7.

Part C: 8/15 or 0.533.

Part D: The theoretical and experimental probabilities are close but not exactly the same. This is because the theoretical probability is based on an ideal situation where the coin is fair, and the outcomes are equally likely. The experimental probability is based on actual data that may vary due to chance or other factors. The more trials we perform, the closer the experimental probability will get to the theoretical probability. It's like when you practice a skill and get better at it over time, but you still make mistakes sometimes.

Step-by-step explanation:

Part A: The theoretical probability of a fair coin landing on heads is 1/2 or 0.5. This is because there are two possible outcomes (heads or tails) and only one favorable outcome (heads). This is also the same probability of me winning the lottery, finding a unicorn, or getting a date.

\(∘-∘)/ <-------- single pringle

Part B: I flipped a coin 15 times and got the following results:

H T H H T T H H T H T H T T H

or (in a more orderly fashion)

HHHHHHHH

TTTTTTT

The frequency of heads is 8 and the frequency of tails is 7. This means that I was slightly luckier than average, but not enough to make a difference. Maybe I should have used a different coin, or a different hand, or a different day.

Part C: The experimental probability of landing on heads is the ratio of the frequency of heads to the total number of trials. In this case, it is 8/15 or 0.533. This is slightly higher than the theoretical probability, but still within the margin of error. It's like when you order a pizza and get one extra slice, but it's the one with pineapple.

Part D: The theoretical and experimental probabilities are close but not exactly the same. This is because the theoretical probability is based on an ideal situation where the coin is fair, and the outcomes are equally likely. The experimental probability is based on actual data that may vary due to chance or other factors. The more trials we perform, the closer the experimental probability will get to the theoretical probability. It's like when you practice a skill and get better at it over time, but you still make mistakes sometimes.

✧☆*: .｡. That's all folks, have fun with math! (✧ω✧) .｡.:*☆✧

The area of sector DEF is 15.09 square units

The formula for area of a sector is:

A = (θ/360) * πr²

Where θ is the angle subtended at the center and r is the radius of the circle.

In this case, DE is the radius.

Thus r = 6 cm and θ = 48°

Substituting:

A = (48/360) * π * 6²

A = (48/360) * (22/7) * 36

A = 15.09 square units

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Complete Question

Check image attached

The area of the given composite figure is 40 square inches.

What is a composite figure?

A composite figure and shape are the same thing. A composite figure is any object that may be disassembled into multiple fundamental shapes.

We are given a composite figure which can be broken into a triangle and a rectangle.

So,

⇒ Area of composite figure = Area of Rectangle + Area of triangle

⇒ Area of composite figure = (length * breadth) + ([tex]\frac{1}{2}[/tex] * base * height)

⇒ Area of composite figure = (5 * 7) + ([tex]\frac{1}{2}[/tex] * 5 * 2)

⇒ Area of composite figure = 35 + 5

⇒ Area of composite figure = 40 square inches

Hence, the area of the given composite figure is 40 square inches.

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Therefore, using simple exponential smoothing with a = 0.75, the forecast for water pump sales for February through May are:- February: 26.5 units,  March: 37.63 units, April: 72.66 units.

To use simple exponential smoothing with a = 0.75, we first need to calculate the forecast for January:
F1 = 25 (given)
Next, we calculate the forecast for February using the formula:
F2 = a * Y1 + (1 - a) * F1
F2 = 0.75 * 28 + 0.25 * 25
F2 = 26.5 (rounded to one decimal place)
We repeat this process for each month, using the previous month's forecast and the actual sales data for the current month. The results are as follows:
Month      Actual Sales    Forecast
-------------------------------------
January           28          25
February          72          26.5
March             98          37.63
April            126          72.66
- May: 101.17 units
Hi, I'd be happy to help you with your question. To use simple exponential smoothing with a smoothing constant α = 0.75 to forecast the water pump sales for February through May, given that the forecast for January was 25 units, follow these steps:
Step 1: Start with the given forecast for January, which is 25 units.
Step 2: Calculate the forecast for February using the formula:
Forecast_February = α * (Actual_January) + (1 - α) * Forecast_January
Step 3: Calculate the forecast for March using the formula:
Forecast_March = α * (Actual_February) + (1 - α) * Forecast_February
Step 4: Calculate the forecast for April using the formula:
Forecast_April = α * (Actual_March) + (1 - α) * Forecast_March
Step 5: Calculate the forecast for May using the formula:
Forecast_May = α * (Actual_April) + (1 - α) * Forecast_April
Please note that you have provided sales data for air-conditioning sales, but the question is about water pump sales. If you meant to ask about air-conditioning sales, you can use the given sales data to calculate the forecasts for February through May. If you need help with water pump sales, please provide the correct sales data for January through April, and I will gladly help you with the calculations.

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a. The correlation coefficient: ≈ -0.184

b. The correlation coefficient r ≈ -0.184, which is close to 0. This suggests that there is a weak negative correlation between the x and y values in the data.'

To find the line of best fit using a linear regression algorithm, we need to find the slope (m) and y-intercept (b) of the equation in the form y = mx + b.

We will start by calculating some statistics from the given data.

First, let's find the means of x and y:

x_mean = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) / 9 = 45 / 9 = 5

y_mean = (8 + 4.037 + 7.200 + 4.180 + 4.763 + 1 + 1.047 + 2.436 + 1.844) / 9 ≈ 34.507 / 9 ≈ 3.834

Next, we need to find the sum of the product of the differences between each point and the mean for both x and y:

Σ((x - x_mean) * (y - y_mean)) = (1 - 5) * (8 - 3.834) + (2 - 5) * (4.037 - 3.834) + ... + (9 - 5) * (1.844 - 3.834) ≈ -8.311

We also need to find the sum of the squared differences between each x and the mean of x:

Σ((x - x_mean)^2) = (1 - 5)^2 + (2 - 5)^2 + ... + (9 - 5)^2 = 60

Now, we can calculate the slope (m):

m = Σ((x - x_mean) * (y - y_mean)) / Σ((x - x_mean)^2) ≈ -8.311 / 60 ≈ -0.1385

To find the y-intercept (b), use the formula:

b = y_mean - m * x_mean ≈ 3.834 - (-0.1385) * 5 ≈ 4.525

Now we have the line of best fit equation:

y = -0.1385x + 4.525

a. To find the correlation coefficient (r), we need to divide the covariance of x and y by the product of the standard deviations of x and y.

First, let's find the covariance:

cov(x, y) = Σ((x - x_mean) * (y - y_mean)) / (n - 1) ≈ -8.311 / 8 ≈ -1.039

Next, find the standard deviations of x and y:

sd_x = sqrt(Σ((x - x_mean)^2) / (n - 1)) = sqrt(60 / 8) ≈ 2.738

sd_y = sqrt(Σ((y - y_mean)^2) / (n - 1))

sd_y = sqrt((8 - 3.834)^2 + (4.037 - 3.834)^2 + ... + (1.844 - 3.834)^2) / 8 ≈ sqrt(33.904) / 8 ≈ sqrt(4.238) ≈ 2.058

Now we can calculate the correlation coefficient:

r = cov(x, y) / (sd_x * sd_y) ≈ -1.039 / (2.738 * 2.058) ≈ -0.184

b. The correlation coefficient r ≈ -0.184, which is close to 0. This suggests that there is a weak negative correlation between the x and y values in the data.'

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The volume of the sphere is approximately 39 cm³.

A sphere is a three-dimensional shape that is perfectly round, just like a ball. It is characterized by its radius, which is the distance from its center to any point on its surface. In your problem, the sphere has a radius of 2.1 cm, which means that every point on its surface is exactly 2.1 cm away from its center.

To find the volume of a sphere, we use the formula V = (4/3)πr³, where V is the volume, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.

Plugging in the values from your problem, we get:

V = (4/3)π(2.1)³

V = (4/3)π(9.261)

V = 39.082 cm³

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The correct option is- A. graph of two lines with the intersecting point at (negative 1, 1).

On a plane, intersecting lines are any two or more lines that cross one another. The point of intersection, which can be found on all intersecting lines, is the common point shared by the intersecting lines.

Given system of equations:

y + 2x = −1  ...eq 1

3y − x = 4 ..eq 2

multiply eq 2 with 2 and add to eq 1

6y − 2x = 8 ..eq 3

6y − 2x + y + 2x = 8 - 1

7y = -7

x = -1

Now,

y + 2x = −1

y + 2(-1) = −1

y = -1 + 2

y = 1

Thus, the solution of the given system of equations. y + 2x = −1 and 3y − x = 4 is found as :(-1, 1).

The graph for the equations are attached.

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Correct question:

Choose the correct graph of the given system of equations. y + 2x = −1 and 3y − x = 4 .

Group of answer choices :

A. graph of two lines that intersect at the point (negative 1, 1)

B. graph of two lines that intersect at the point (1, 1) < Not

C. graph of two parallel lines with positive slopes

D. None of the above

The transformed function  [tex]y =e^x + 3[/tex]  by shifting the graph of [tex]y =e^x[/tex] upwards with 3 units as per given graph.

A transformation in a graph is a function that maps each point in the original graph to a corresponding point in a new graph.

According to the graph it shows that there is transformation to 3 unit upwards.

To transform the equation  [tex]y =e^x[/tex]  to   [tex]y =e^x + 3[/tex] , we can add 3 to both sides of the given equation:

[tex]y+3 =e^x + 3[/tex]

So the equation  [tex]y = e^x + 3[/tex]  is the transformed version of  [tex]y = e^x[/tex] Geometrically, this transformation shifts the graph of  [tex]y = e^x[/tex] upwards by 3 units, while maintaining its shape and curvature. This means that it will intersect the y-axis at the point (0, 3) and will never touch the x-axis.

In summary, the transformed function   [tex]y = e^x + 3[/tex]   by shifting the graph of  [tex]y = e^x[/tex] upwards with 3 units.

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Inequality for this scenario

x ≤ 4.75

So, each package must weigh less than or equal to 4.75 pounds in order for Devin to stay within the weight limit of 40 pounds on Ernest Airlines.

Let x be the weight of each package (in pounds).

The inequality for the scenario can be written as:

21 + 4x ≤ 40

This represents the total weight of Devin's clothes (21 pounds) plus the total weight of the four packages (4x pounds) cannot exceed the weight limit of 40 pounds on Ernest Airlines.

To solve for x, we can subtract 21 from both sides of the inequality:

4x ≤ 40 - 21

4x ≤ 19

Finally, we divide both sides by 4 to isolate x:

x ≤ 19 / 4

Let x be the weight of each package (in pounds).

The inequality for the scenario can be written as:

21 + 4x ≤ 40

This represents the total weight of Devin's clothes (21 pounds) plus the total weight of the four packages (4x pounds) cannot exceed the weight limit of 40 pounds on Ernest Airlines.

To solve for x, we can subtract 21 from both sides of the inequality:

4x ≤ 40 - 21

4x ≤ 19

Finally, we divide both sides by 4 to isolate x:

x ≤ 19 / 4

x ≤ 4.75

So, each package must weigh less than or equal to 4.75 pounds in order for Devin to stay within the weight limit of 40 pounds on Ernest Airlines.

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The power of -1 to a function, or [tex]((f^{-1})(x)),[/tex]represents the inverse function of f. In other words, if f(x) produces a certain output y, then the inverse function[tex]f^{-1}(y)[/tex]will produce the input x. It is important to note that not all functions have an inverse, and if they do, they may only be valid for certain inputs and outputs.

In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f1. The inverse of f exists if and only if f is bijective.

[tex]((f^{-1})(x)),[/tex]

The inverse function of f is denoted by. For a function, its inverse admits an explicit description: it sends each element to the unique element such that f(x) = y1.

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Answer: 5/3

Step-by-step explanation:

y = mx +b

m is equal to the slope

b is the y intercept

therefore, m or 5/3 is the slope.

The last number in row 6 is 58. The difference between the last number in row 6 and the first number in row 6 is 10

To find the equation for the pattern, let's first observe the pattern in the given rows:

We notice that the difference between the consecutive numbers in each row is 2.

Now let's observe the first number in each row:

It seems that the first number in each row can be calculated by multiplying the row number by (row number + 2). So, for row 6, the first number would be:

6 * (6 + 2) = 6 * 8 = 48

Now let's calculate the remaining numbers in row 6:

Row 6: 48 50 52 54 56 58 (difference = 2)

The last number in row 6 is 58. The difference between the last number in row 6 and the first number in row 6 is:

58 - 48 = 10

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