Mathematics College

For each triangle, calculate the ratio of the opposite side to the adiacent side to two decimal places. How do they compare?

Answers

Answer 1

Answer:

[tex]Triangle\: 1: } \dfrac{opposite}{adjacent} = \dfrac{16}{39.6} = \boxed{0.40404\cdots}[/tex]

[tex]Triangle\: 2: } \dfrac{opposite}{adjacent} = \dfrac{48}{118.8} = \boxed{0.40404\cdots}[/tex]

Choice C: The ratio is the same for both triangles

Step-by-step explanation:

[tex]Triangle\: 1: } \dfrac{opposite}{adjacent} = \dfrac{16}{39.6} = \boxed{0.40404\cdots}[/tex]

[tex]Triangle\: 2: } \dfrac{opposite}{adjacent} = \dfrac{48}{118.8} = \boxed{0.40404\cdots}[/tex]

So both ratios are the same. This is also because the ratios represent the tan of the angle that the adjacent side makes with the hypotenuse and this angle of 22° is same in both triangles

Related Questions

Based on the graph below, what will be the coordinates of the point that represents the number of snowballs needed to make 2 snowmen?

Answers

the coordinates of the point that represents the number of snowballs needed to make 2 snowmen should be option C. which is (2,6)

Use a graphing calculator to sketch the graph of the quadratic equation, and then state the domain and range.
y = negative 0.5 x squared + 0.7 x minus 5.1
a.
D: all real numbers
R: (y less-than-or-equal-to negative 5.1)
c.
D: all real numbers
R: (y less-than-or-equal-to negative 4.855)
b.
D: all real numbers
R: (y greater-than-or-equal-to 4.855)
d.
D: all real numbers
R: (y less-than-or-equal-to negative 5.345)

Please select the best answer from the choices provided

A
B
C
D

Answers

The domain is all real numbers, and the range is (y ≤ -4.855). (option b)

To sketch the graph of a quadratic equation, you can use a graphing calculator, which is a handy tool that can produce a visual representation of the equation. For instance, let's take the quadratic equation y = -0.5x² + 0.7x - 5.1 and sketch its graph using a graphing calculator.

When we enter this equation into the graphing calculator, we can see a U-shaped curve that is symmetric about the vertical line passing through the vertex. The vertex is the point where the parabola changes direction and is given by the formula x = -b/2a, y = f(x), where f(x) is the value of y at the vertex.

Now let's examine the given options and determine their domain and range based on the graph of the quadratic equation.

all real numbers, R: (y ≤ -4.855)

For this option, we can observe that the parabola opens upward, and the vertex is at (0.7, -5.35).

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Why do polling companies often survey 1060 individuals when they wish to estimate a population proportion with a margin of error 3% with 95% confidence?

Answers

Polling companies survey 1060 individuals to estimate a population proportion with a margin of error of 3% with 95% confidence because it strikes a balance between cost and accuracy, and it is based on statistical calculations that aim to achieve a reasonable level of precision while minimizing the margin of error.

When a polling company selects a sample size of 1060 individuals, it is based on statistical calculations that aim to achieve a 3% margin of error with 95% confidence. The sample size needs to be large enough to reduce the margin of error and increase the level of confidence in the results.

The margin of error decreases as the sample size increases, and it is inversely proportional to the square root of the sample size. Therefore, a larger sample size leads to a smaller margin of error. However, increasing the sample size also increases the cost and time required to conduct the survey.

Polling companies aim to strike a balance between the cost and the desired level of accuracy, which is typically a margin of error of 3-5% with 95% confidence. A sample size of 1060 individuals is often considered a reasonable compromise between cost and accuracy, as it provides a small enough margin of error to be useful while being feasible to conduct within a reasonable time and budget.

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Answer the question below

Answers

Answer:

B represents a function.

The graph of y=g(x) is shown. Draw the graph of y=-g(x)+1

Answers

The graph of the absolute value function y = -g(x) + 1 is shown in the image attached below.

The graph of the absolute value function y = 2h(x + 4) is shown in the image attached below.

What is a translation?

In Mathematics, the translation a geometric figure or graph upward simply means adding a digit to the value on the y-coordinate of the pre-image or function.

In Mathematics, a vertical translation to the positive y-direction (upward) is modeled by this mathematical expression g(x) = f(x) + N.

Where:

N represents an integer.g(x) and f(x) represent a function.

In this scenario, we can logically deduce that the equation of the first parent absolute value function is g(x) = |2x| and it would be reflected over the y-axis, and then translated by 1 units upward;

g(x) = |2x|

y = -g(x) + 1 = |2x| + 1

h(x) = |2x + 4| - 4

y = 2h(x + 4)

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Decimal word problem
pls help
Can someone give me a decimal word problem that involves subtraction and division plssss with the answer step by stepp plsss help i begg u guys lovee uuu guyss

Answers

Thus, decimal word problem that involves subtraction and division-

The two empty water container weighs 0.64 kg and 1.728 kg when filled with water. Find the water weigh?

Explain about the decimal word problem:

The decimal point, which distinguishes between a decimal number's whole number and fractional component, is a little dot. Decimal numbers include 4.14, 3.786, 5.453, and 123.5687 as examples.

Any number of digits can be used to represent the decimal places in a decimal number.The decimal places are the digits that come after the decimal point and are arranged in the following order: tenths, hundredths, thousandths, ten-thousandths, and so forth.

decimal word problem : (subtraction and division)

The two empty water container weighs 0.64 kg and 1.728 kg when filled with water. Find the water weigh?

2 water pitcher's weight when empty and full with water is 0.64 kg and 1.728 kg, respectively. We must determine the water's weight. We must deduct the weight of the container when it is empty from the weight of a container when it is full of water in order to determine the weight of the water.

Division: Weight of one water container = 0.64 / 2 = 0.34 kg.

Subtraction: Weight of water : 0.728 - 0.34 = 0.385 kg.

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The stem-and-leaf plot displays the amount of time, in minutes, that 10 players spent practicing their serving skills for an upcoming tennis tournament.

1 2, 2, 9
2 0, 2, 6
3 1, 1
4 0
5
6 8
Key: 3|1
means 31

Part A: Calculate the mean, median, mode, and range for the data given. Show your work. (8 points)

Part B: Should the tennis players report the mean or median value to show they are less prepared for the tournament? Explain based on the scenario. (4 points)

Answers

Part A:

To calculate the mean, we add up all the values and divide by the total number of values:

12 + 12 + 19 + 20 + 22 + 26 + 31 + 31 + 40 + 68 = 281

Mean = 281 / 10 = 28.1

To find the median, we need to find the middle value. Since there is an even number of values, we take the average of the two middle values:

Median = (22 + 26) / 2 = 24

The mode is the value that appears most frequently. In this case, there is no mode as no value appears more than once.

The range is the difference between the highest and lowest values:

Range = 68 - 12 = 56

Part B:

The median may be a better measure of central tendency in this scenario as there is an outlier value of 68 that can skew the mean. Since outliers can have a significant impact on the mean, reporting the median may give a better representation of the typical amount of time spent practicing for the tournament. Therefore, the tennis players should report the median value.

Five friends-Allison, Beth, Carol, Diane, and Evelyn- have identical calculators and are studying for a statistics exam. They set their calculators down in a pile before taking a study break and then pick them up in random order when they return from the break. What is the probability that at least one of the five gets her own calculator? [Hint: Let A be the event that Alice gets her own calculator, and define events B, C, D, and E analogously for the other four stu- dents.] How can the event (at least one gets her own calcu- lator} be expressed in terms of the five events A, B, C, D, and E? Now use a general law of probability. [Note: This is called the matching problem. Its solution is easily extended to individuals. Can you recognize the result when n is large (the approximation to the resulting series)?]

Answers

The probability that at least one of the five friends gets her own calculator can be expressed in terms of the events A, B, C, D, and E as follows:

P(at least one gets her own calculator) = 1 - P(none of them get their own calculator)

To calculate P(none of them get their own calculator), we can use the principle of inclusion-exclusion:

P(none of them get their own calculator) = 5!/(5-5)!*(0!)- 4!/(5-4)!*(1!)+ 3!/(5-3)!*(2!)- 2!/(5-2)!*(3!)+ 1!/(5-1)!*(4!)
= 1 - 1 + 10/120 - 15/120 + 24/120
= 8/15

Therefore, the probability that at least one of the five friends gets her own calculator is:

P(at least one gets her own calculator) = 1 - P(none of them get their own calculator)
= 1 - 8/15
= 7/15

Verify the following Pythagorean identity for all values of x and y:
(x² + y²)² = (x² - y²)² + (2xy)²

Answers

The identity (x² + y²)² = (x² - y²)² + (2xy)²

What is an identity?

An identity is a mathematical equation which shows that one side of the equation is equivalent and equal to the other side of the equation.

To verify the following Pythagorean identity for all values of x and y:

(x² + y²)² = (x² - y²)² + (2xy)², we need to shows that

left-hand side L.H.S = right hand side

So, we proceed as follows

L.H.S = (x² + y²)²

Expanding the brackets, we have that

(x² + y²)² = (x² + y²)(x² + y²)

= (x²)² + 2x²y² + (y²)²

= (x²)² + (y²)² + 2x²y²

Now, we know that (a - b)² = a² + b² - 2ab. So,

a² + b² = (a - b)² + 2ab

Let a = x² and b = y²

So, we have that

(x²)² + (y²)² = (x² - y²)² + 2x²y²

So, substituting this into the equation, we have that

(x²)² + (y²)² + 2x²y² = (x² - y²)² + 2x²y² + 2x²y²

= (x² - y²)² + 4x²y²

= (x² - y²)² + 2²x²y²

= (x² - y²)² + (2xy)²

So, Since L.H.S = R.H.S

(x² + y²)² = (x² - y²)² + (2xy)²

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Draw fraction strips to show the following fractions: 4/6,1/6, and 5/6. Then write the three fractions in order from least to greatest.

Answers

Here are the fraction strips for 4/6, 1/6, and 5/6:

```
4/6: | | | | / / /
1/6: | / / / / / /
5/6: | | | | | / /

```

Writing the fractions in order from least to greatest:

1/6, 4/6, 5/6.

Please help me with these 6th grade math questions. Thank you so much!

Answers

The number line is shown below:

How to solve

The given inequality:

x -5 < 11

Add 5 on both sides

x -5 + 5 < 11 + 5

Add x < 16

Thus, this is represented in the number line below:

A graphical illustration of numbers in a line is known as a number line. It is utilized for indicating the relation between values, and also to do calculations such as subtraction, multiplication, division, etc. The figures get bigger from left to right and diminishes from right to left.

Answers

Answer:

Profit when producing 70 items = $3,050

Profit when producing 80 items = $1,350

Step-by-step explanation:

To find the profit when producing a certain number of items, we need to subtract the cost from the revenue:

Profit = Revenue - Cost

Let's calculate the profit for producing 70 items:

Revenue when producing 70 items:

R(70) = -0.5(70-120)^2 + 7,200 = $6,800

Cost of producing 70 items:

C(70) = 50(70) + 250 = $3,750

Profit when producing 70 items:

Profit = Revenue - Cost = $6,800 - $3,750 = $3,050

Similarly, let's calculate the profit for producing 80 items:

Revenue when producing 80 items:

R(80) = -0.5(80-120)^2 + 7,200 = $5,600

Cost of producing 80 items:

C(80) = 50(80) + 250 = $4,250

Profit when producing 80 items:

Profit = Revenue - Cost = $5,600 - $4,250 = $1,350

We can see that the profit is less when producing 10 more units because the cost of producing those additional units exceeds the revenue generated from selling them. In other words, the marginal cost (the cost of producing one additional unit) is greater than the marginal revenue (the revenue generated from selling one additional unit) beyond a certain point.

This is an example of the law of diminishing returns, which states that as we increase the quantity of inputs while keeping other inputs constant, the marginal product (output per unit of input) eventually decreases.

Hope this helps!

Evaluate each expression (a) log7 1/7 =
(b) log 3 27 = 9

Answers

(a) log7 1/7 = -1

(b) log3 27 = 3

Explanation:

(a) The logarithm of 1/7 to the base 7 is -1. This is because 7 raised to the power of -1 equals 1/7. Therefore, log7 1/7 = -1.

(b) The logarithm of 27 to the base 3 is 3. This is because 3 raised to the power of 3 equals 27. Therefore, log3 27 = 3.

Which two lanes are parallel

1. p || q
2. r || s
3. q || r
4. p || r

Answers

Answer:

Step-by-step explanation:

1p:80°-140。＝40°，40°

1. Line p is parallel to line q
In the figure, we see that line p crosses transversal n at 140 degrees. This means that it’s supplementary angle must be 40 degrees, as 140 and 40 add to 180. The supplementary angle on line p is congruent to the angle in line q, which we know to be 40 degrees. Therefore, p and q cross the transversal at the same angle, therefore they are parallel.

Triangle P is rotated 180° about (0, 0) to give triangle Q.
Triangle Q is translated by 5 -2
to give triangle R.
(a) Describe fully the single transformation that maps triangle P onto triangle R.

Answers

Answer:

The single transformation that maps triangle P onto triangle R is a reflection about the origin followed by a translation of 5 units to the right and 2 units down.

Johnny’s Donut Shop is running a “buy one get one 50% off” sale today. During the sale, Laine bought 4 glazed donuts for only $3.78. What is the regular price for one glazed donut at Johnny’s Donut Shop?

Answers

Answer:

Step-by-step explanation:

Set up a simple system.

Price = 4x where x = price of donut

For the sale, the price = 3.78, but the x value is 0.5x as it is half off.

$3.78 = 4(0.5x)

Solve for x =$1.89

Therefore, the original price of the donuts were $1.89.

Check by doing 4 x 50%(.5) of 1.89 which equals 3.78!

a school needed a total of $3,546 in Supplies for grade 3,4 and 5 if they spilt the supplies evenly how much did they spend on each grade?

Answers

Answer: $1.182

Step-by-step explanation: 3.546 divided by 3

Answer:

$1,182

Step-by-step explanation:

I just took the total of $3546 and divided the total by the three grade levels so 3 and I got the answer 3,546. But honestly i'm not sure if it's right lol, so if it's incorrect let me know and I'll work into it more.

2(x-1)+3(x+1)=4(x+4)

Answers

1. Distribute. Everything. 2x - 2 + 3x + 3 = 4x + 16.
2. Add your like terms. (must be on the same side) 5x + 1 = 4x + 16
3. Single out x by subtracting 1. 5x = 4x + 16 - 1.
4. Simplify. 5x = 4x + 15
5. Subtract 4x from both sides to combine like terms. 5x - 4x = 15.
6. Simplify. 1x = 15.

That means your final answer would be x = 15. If you were to input 15 into every x, it would simplify to 76 = 76, which means x is definitely 15.

Answer:

x = 15

Step-by-step explanation:

Simplifying the expression:

2(x - 1) + 3(x + 1) = 4(x + 4)2x - 2 + 3(x + 1) = 4(x + 4)2x - 2 + 3(x + 1) = 4(x + 4)2x - 2 + 3x + 3 = 4(x + 4)2x - 2 + 3x + 3 = 4(x + 4)2x - 2 + 3x + 3 = 4x + 16

Add the numbers:

2x - 2 + 3x + 3 = 4x + 162x + 3x - 2 + 3 = 4x + 162x + 3x + 1 = 4x + 16

Combine like terms:

2x + 3x + 1 = 4x + 165x + 1 = 4x + 16

Subtract 1 from both sides:

5x + 1 - 1 = 4x + 16 - 15x = 4x + 15

Subtract 4x from both sides:

5x - 4x = 4x - 4x + 15x = 15

Answer:

x = 15

Step-by-step explanation:

Simplifying the expression:

2(x - 1) + 3(x + 1) = 4(x + 4)2x - 2 + 3(x + 1) = 4(x + 4)2x - 2 + 3(x + 1) = 4(x + 4)2x - 2 + 3x + 3 = 4(x + 4)2x - 2 + 3x + 3 = 4(x + 4)2x - 2 + 3x + 3 = 4x + 16

Add the numbers:

2x - 2 + 3x + 3 = 4x + 162x + 3x - 2 + 3 = 4x + 162x + 3x + 1 = 4x + 16

Combine like terms:

2x + 3x + 1 = 4x + 165x + 1 = 4x + 16

Subtract 1 from both sides:

5x + 1 - 1 = 4x + 16 - 15x = 4x + 15

Subtract 4x from both sides:

5x - 4x = 4x - 4x + 15x = 15

Given this equation what is the value of y at the indicated point?

Answers

We are given the equation of a parabola, [tex]y=-x^2+2[/tex]. The question asks us to find both values of a point given the the x-coordinate which is -3. To find the y-coordinate, simply plug in -3 for x into the equation of the parabola.

[tex]y=-x^2+2 \Longrightarrow y=-(-3)^2+2 \Longrightarrow y=-(9)+2 \Longrightarrow y=-9+2[/tex]

[tex]\Longrightarrow y=-7[/tex]

Thus, the point is (-3,-7)

4. Gabriel wants to learn how to play darts, so
he researches the costs of different dartboards
at stores in his area. What would be the most
appropriate way for him to display his data?
Answer will be marked!!

Answers

Answer:

a line graph

Step-by-step explanation:

Lauren over-filled the homemade pecan pie that she was baking for Thanksgiving, so the pie needed additional cooking time. Lauren decided to place a strip of aluminum foil around the edge of the crust so that it would not burn. If Lauren used a pie pan with a 12-inch diameter, how long, to the nearest inch, should the strip of foil be?

A. 19 inches
B. 24 inchess
C. 113 inchess
D. 38 inchess

Answers

Hence the correct option is D. 38inches long, to the nearest inch, should the strip of foil be.

What is the circumference?

The circumference of a circle in mathematics is its perimeter. That is, if the circle were opened up and straightened out to a line segment, the circumference would be the length of the arc. In according to broader the perimeter is the length of any closed figure of curve.

What is diameter?

A straight line and cuts through the middle of a body or figure. particularly the length of a diameter is a line segment passing through the center of a circle with its ends on the circumference.

We determine the circumference of the pie pan in order to determine the length of the aluminium foil strip required.

Circumference =2[tex]\pi[/tex]r.

where, r = circle of radius and [tex]\pi[/tex] = mathematical constant =3.14.

The pie pan of radius = half of its diameter=12 inches/2= 6 inches.

Now, the pie pan of circumference is:

Circumference: 2 [tex]\pi[/tex]r=2[tex]\pi[/tex] *6 inches= 37.7 inches

Therefore , the length of the aluminium foil strip = 38 inches,

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Experiment 4: You have a bag of 15 scrabble tiles. 3 of the
tiles are the letter O. And the other 2 tiles are the letter U.
a. What is the probability of getting an O on the first draw?
b. What is the probability of getting a U if you didn't replace the O?
What is the probability of getting O and then U if you don't
replace the O?
c.

Answers

Answer:

a. The probability of getting an O on the first draw can be calculated by dividing the number of O tiles by the total number of tiles in the bag:

```

P(O on first draw) = 3/15 = 0.2

```

Therefore, the probability of getting an O on the first draw is 0.2 or 20%.

b. After removing one O tile, there are 14 tiles left in the bag, of which 2 are U tiles. Therefore, the probability of getting a U if you didn't replace the O is:

```

P(U without replacing O) = 2/14 = 1/7 ≈ 0.143

```

Therefore, the probability of getting a U if you didn't replace the O is approximately 0.143 or 14.3%.

c. The probability of getting O and then U if you don't replace the O can be calculated as follows:

```

P(O and then U) = P(O on first draw) * P(U without replacing O)

= 3/15 * 2/14

= 1/35

≈ 0.029

```

Therefore, the probability of getting O and then U if you don't replace the O is approximately 0.029 or 2.9%.

Step-by-step explanation:

How does this work lol?

Answers

Answer:

this is the Pythagorean theorem this is the formula a^2 + b^2 = c^2

Step-by-step explanation:

please mark as brainliest have a great day :D

a^2+b^2=c^2 that’s the answer to your question I hope I helped you

Can y’all please help me with this whole thing :/

Answers

Answer:

Slope-Intercept Form: y=2x+1

Standard Form: 2x-y=6

1. [tex]y=\frac{1}{3}x-1[/tex]

2. [tex]y=-\frac{2}{5}x[/tex]

3. [tex]y=x+8[/tex]

4. [tex]y=-\frac{1}{2}x+3[/tex]

5. [tex]y=2x+1[/tex]

6. [tex]y=-4x-5[/tex]

7. [tex]y=-x+2[/tex]

8. [tex]y=\frac{2}{3}x-4[/tex]

9. [tex]y=x\\[/tex]

Triangle NMO has vertices at N(−5, 2), M(−2, 1), O(−3 , 3). Determine the vertices of image N′M′O′ if the preimage is reflected over x = −2. N′(1, 2), M′(−2, 1), O′(−1, 3) N′(−5, −6), M′(−2, −5), O′(−3, −7) N′(−5, 0), M′(−2, −1), O′(−3, 1) N′(9, 2), M′(6, 1), O′(7, 3)

Answers

The vertices of image N′M′O′ if the preimage is reflected over x = −2 include the following: D. N′(9, 2), M′(6, 1), O′(7, 3)

What is a reflection over the y-axis?

In Mathematics and Geometry, a reflection over or across the y-axis or line x = 0 is represented and modeled by this transformation rule (x, y) → (-x, y).

Similarly, the reflection of a point (x, y) over the line x = a is modeled by this transformation rule:

(x, y) → (-2a - x, y)

(x, y) → (-2(-2) - x, y)

(x, y) → (4a - x, y)

By applying a reflection over x = -2 to the coordinate of the given triangle NMO, we have the following coordinates:

(x, y) → (4 - x, y).

N (−5, 2) → (9, 2)

M(−2, 1) → (6, 1)

O(−3 , 3) → (7, 3)

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Answer:

N′(−5, −6), M′(−2, −5), O′(−3, −7)

Step-by-step explanation:

I am in the middle of taking the test and even made a graph to check my work! This should be the correct answer

If you drive from Dublin to Galway at a constant speed of 70 miles per hour, but pause
for 30 minutes for lunch, how long is your trip in total? Round your final answer to the
nearest tenth of an hour

Answers

0 miles and hour suiii

The distance between Dublin and Galway is approximately 125 miles.

If you drive at a constant speed of 70 miles per hour, it will take you 125 / 70 = 1.79 hours to get to Galway.

Pausing for 30 minutes for lunch means you will add 0.5 hours to your trip.

So, your trip in total will be 1.79 + 0.5 = 2.29 hours.

Rounding to the nearest tenth of an hour, your trip will take 2.3 hours in total.

Convert the rectangular coordinates to polar coordinates with
r > 0 and 0 ≤ < 2.
(1, −2)
Help I'm stuck with this one :(

Answers

If you want to transform rectangular coordinates (x, y) into its polar counterpart (r, θ), do the following:

The Steps

Firstly, obtain r by computing the distance from the origin to point (x,y), making use of the Pythagorean theorem or simply put, by evaluating r = sqrt(x^2 + y^2).

Secondly, acquire θ which denotes the angle formed between the positive x-axis and a line correlating the origin and the coordinate (x,y). Calculate this via inverse tangent function: θ = atan2(y, x).

Thirdly, in case the measurement for θ is indicated in radians while you require degrees instead, multiply by 180/π to convert it.

Usually expressed as (r, θ), where r indicates proximity disentangled between the origin and the point (x,y), whilst θ identifies the aforementioned angle.

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based on the value of r^2, which modeling equation is best fit for the data set? a. y=16.5839•1.0185^x
b. y=0.985515x-2.81333
c. y=-0.0000758936x^3+0.0143444x^2+0.207549x+7.70667
d. y=0.00182197x^2+0.785098x+1.195

Answers

Okay, let's analyze the r^2 values for the 4 options:

a. y=16.5839•1.0185^x

r^2 is undefined, as this is an exponential model. We cannot determine how well it fits the data based only on r^2.

b. y=0.985515x-2.81333

This is a linear model, so r^2 would be between 0 and 1. Without knowing the actual r^2 value, we cannot determine how good the fit is.

c. y=-0.0000758936x^3+0.0143444x^2+0.207549x+7.70667

This is a 3rd order polynomial model. Again, without the r^2 value, we cannot determine the quality of fit. It could be a good or poor fit.

d. y=0.00182197x^2+0.785098x+1.195

If this model has the highest r^2 value (closest to 1), it would indicate the best fit to the data of the options.

In summary, without seeing the actual r^2 values calculated from the data for each model, we cannot definitively say which option is the "best fit". The model with the r^2 closest to 1 would likely be the best, but r^2 only tells part of the story. Other factors like complexity, interpretability, and residual analysis also matter. But based solely on r^2, the quadratic model d seems the most likely to be the best fit, as long as its r^2 value is high.

Does this help explain the approach? Let me know if you have any other questions!

Find the equation of the plane passing through the lines of intersection of the planes :
2x - 7y + 4z = 3 , 3x - 5y + 4z + 11 = 0 and the point ( -2 , 1 , 3 ).

Answers

The equation of the plane passing through the lines of intersection of the planes 2x - 7y + 4z = 3 and 3x - 5y + 4z + 11 = 0 and the point (-2, 1, 3) is

-13x - 8y + z + 90 = 0

We have,

To find the equation of the plane passing through the intersection of two planes, we need to first find the direction vector of the line of intersection of the two planes.

Let's call the two planes P1 and P2, which are given by the equations:

P1: 2x - 7y + 4z = 3

P2: 3x - 5y + 4z + 11 = 0

To find the direction vector of the line of intersection of P1 and P2, we can take the cross product of the normal vectors of P1 and P2. The normal vectors are:

n1 = <2, -7, 4>

n2 = <3, -5, 4>

Taking the cross-product of n1 and n2.

n1 x n2 = <-13, -8, 1>

This vector is parallel to the line of intersection of P1 and P2. We can use this vector as the direction vector for the line.

Now, we need to find a point on the line.

We can use the point of intersection of P1 and P2 as a point on the line.

To find this point, we can solve the system of equations:

2x - 7y + 4z = 3

3x - 5y + 4z + 11 = 0

Subtracting the first equation from the second, we get:

x + 2y + 7 = 0

Solving for x in terms of y,.

x = -2y - 7

Substituting this expression for x into the first equation, we get:

-4y - 14 - 7y + 4z = 3

Simplifying.

-11y + 4z = 17

Choosing y = 1.

-11 + 4z = 17

Solving for z.

z = 7

Substituting y = 1 and z = 7 into the equation for x.

x = -2(1) - 7 = -9

So a point on the line is (-9, 1, 7).

Now, we can use the point (-9, 1, 7) and the direction vector <-13, -8, 1> to write the equation of the plane passing through the line of intersection of P1 and P2.

Let's call this plane P3.

Using the point-normal form of the equation of a plane.

-13(x + 9) - 8(y - 1) + (z - 7) = 0

Expanding and simplifying, we get:

-13x - 8y + z + 90 = 0

Thus,

The equation of the plane passing through the lines of intersection of the planes 2x - 7y + 4z = 3 and 3x - 5y + 4z + 11 = 0 and the point (-2, 1, 3) is:

-13x - 8y + z + 90 = 0

Learn more about equations here:

https://brainly.com/question/17194269

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Plot - 1 1/4 and 5/2 on the number line below.

Answers

The number line where we plotted 1 1/4 and 5/2 is added as an attachment

Plotting -1 1/4 and 5/2 on a number line

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

-1 1/4 and 5/2

To start with, we convert both numbers to the same form

i.e. decimal or fraction

When converted to fractions, we have

-5/4 and 5/2

Rewrite the denominators of the fractions

So, we have

-5/4 and 10/4

This means that we can plot -5/4 at -5 and 10/4 at point 10 where the difference in each interval is 1/4

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

The number line is attached