Find the area of each face of the triangular prism and the total surface area.
Drag each tile to its matching measurement. Each number may be used once or not at all.

The points that are solutions to the equation [tex]2x^2 + y = 4[/tex] are:

I. (3, -14)

III. (-3, -14)

To determine which of the given points are solutions to the equation [tex]2x^2 + y = 4[/tex], we need to substitute the x and y values of each point into the equation and check if the equation holds true.

Let's evaluate each point one by one:

I. (3, -14)

Substituting x = 3 and y = -14 into the equation:

[tex]2(3)^2 + (-14) = 4[/tex]

18 - 14 = 4

4 = 4

Since both sides of the equation are equal, the point (3, -14) is a solution to the equation.

II. (-3, 14)

Substituting x = -3 and y = 14 into the equation:

[tex]2(-3)^2 + 14 = 4[/tex]

18 + 14 = 4

32 = 4

Since the equation is not satisfied (32 is not equal to 4), the point (-3, 14) is not a solution to the equation.

III. (-3, -14)

Substituting x = -3 and y = -14 into the equation:

[tex]2(-3)^2 + (-14) = 4[/tex]

18 - 14 = 4

4 = 4

Since both sides of the equation are equal, the point (-3, -14) is a solution to the equation.

In summary, the points that are solutions to the equation [tex]2x^2 + y = 4[/tex]are:

I. (3, -14)

III. (-3, -14)

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Answer:

The fraction of cupcakes they ate is:

5 / 8

The fraction of cupcakes left is:

3 / 8

Given statement solution is :- The slope (m) of the line is -2, and the y-intercept (b) is -5/4.

To find the y-intercept and slope of the line represented by the equation -8x - 4y = 5, we need to rearrange the equation into slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept.

Let's begin by isolating the term with y:

-8x - 4y = 5

Subtract -8x from both sides:

-4y = 8x + 5

Next, divide both sides of the equation by -4 to solve for y:

y = (-8/4)x - (5/4)

Simplifying further:

y = -2x - 5/4

Now we can identify the slope and the y-intercept:

The slope (m) of the line is -2, and the y-intercept (b) is -5/4.

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The solution is; C. x² + 2x - 1 = 3 equation could be solved using this application of the quadratic formula.

Here,

Quadratic formula: x = -b±√b²-4ac/2a     [ +/-   is   ± ]    

You can find the value of a, b, c  -->  ax² + bx + c = 0  

we have,

x = -2±√2² - 4×1× -4 / 2×1    

Since this is not simplified, you can find a, b, c:

a = 1

b = 2

c = -4

A.)  x² + 1 = 2x − 3   Make the equation into ax² + bx + c = 0.

Subtract 2x on both sides, and add 3 on both sides to set the equation equal to 0

x² + 1 - 2x + 3 = 2x - 2x - 3 + 3

x² - 2x + 4 = 0      

a = 1

b = -2

c = 4    This is not the answer because b = 2 not -2, and c = -4 not 4

B.) x² - 2x − 1 = 3     Subtract 3 on both sides to set the equation = 0

x² - 2x - 4 = 0

a = 1

b = -2

c = -4      This is not the answer because b = 2 not -2

C. x² + 2x - 1 = 3    Subtract 3 on both sides to set the equation = 0

x² + 2x - 4 = 0

a = 1

b = 2

c = -4    This is your answer

D. x² + 2x - 1 = -3      Add 3 on both sides to set the equation = 0

x² + 2x + 2 = 0

a = 1

b = 2

c = 2    

This is not the answer because c = -4 not 2

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complete question:

Which equation could be solved using this application of the quadratic formula?

x = -2±√2² - 4×1× -4 / 2×1    

I did some research and found that for quadratic equations in the form of [tex]y = ax^2 + bx + c[/tex], you can find the minimum value using the equation [tex]$\text{minimum} = c - \frac{b^2}{4a}[/tex] .

We can substitute our given values in the formula and get the following: [tex]$37 = 118 - \frac{b^2}{4}$[/tex] .

We can go ahead and solve it, and get the following:

[tex]-81 = -\frac{b^{2}}{4}[/tex]

[tex]324 = b^2[/tex]

[tex]\pm{18} = b[/tex]

Now we know that the equation is either [tex]$f(x)=x^2+18x+118$[/tex]  or [tex]$f(x)=x^2-18x+118$[/tex] .

We will solve using both equations, but we will solve the one with a positive b-value first. Substituting [tex]f(x)[/tex] for [tex]$37$[/tex], the minimum or y-value of the vertex, we can now solve for the x-value of the vertex.

We have:

[tex]37 = x^2+18x+118[/tex]

[tex]0=x^2+18x+81[/tex]

[tex]0=(x+9)^2[/tex]

[tex]x+9=0\\x=-9[/tex]

Doing the same thing with the second equation, we get:

[tex]37 = x^2-18x+118[/tex]

[tex]0=x^2-18x+81[/tex]

[tex]0=(x-9)^2[/tex]

[tex]x-9=0\\x=9[/tex]

After all of this, we know our vertex's x-values are either [tex]9[/tex] or [tex]-9[/tex], and that our y-value is [tex]37\\[/tex].

We can conclude that our k-value (y-value of the vertex in vertex form) is [tex]37[/tex], and our h-value (x-value of the vertex in vertex form) is [tex]9[/tex] or [tex]-9[/tex].

Conclusion:         [tex]h = -9, 9[/tex]

                            [tex]k=37[/tex]

Answer: 0

Step-by-step explanation:

To solve the equation (x² + 3x)³ - 16(x² + 3x) - 36 = 0 using substitution, let's make a substitution:

Let u = x² + 3x.

Now, we can rewrite the equation in terms of u:

u³ - 16u - 36 = 0.

Let's solve this equation for u by factoring:

(u - 6)(u² + 6u + 6) = 0.

Now, we have two possible cases:

Case 1: u - 6 = 0.

This gives us u = 6.

Case 2: u² + 6u + 6 = 0.

To solve this quadratic equation, we can use the quadratic formula:

u = (-b ± √(b² - 4ac)) / (2a),

where a = 1, b = 6, and c = 6.

Plugging in these values, we get:

u = (-6 ± √(6² - 4(1)(6))) / (2(1)),

u = (-6 ± √(36 - 24)) / 2,

u = (-6 ± √12) / 2,

u = (-6 ± 2√3) / 2,

u = -3 ± √3.

Now that we have the possible values of u, let's substitute back to find the corresponding values of x:

Case 1: u = 6.

Since u = x² + 3x, we have x² + 3x = 6.

Rearranging, we get x² + 3x - 6 = 0.

Hope it helps!

Answer:

A = 30 cm

B = 14 cm

C = 15 cm

D = 41 cm

Step-by-step explanation:

You can find A by multiplying 10 times 6 (because the other side of the square is ten and the side of A already has that 4 there it would be 6) which would be 60, then since it is a right triangle, you can divide that by two and get 30.

You can find B by multiplying 4 by 7 (because the other part of that side of the square is 3 so that part would be 7) which would be 28, then since it is a right triangle, you can divide that by two and get 14.

You can find C by multiplying 10 times 3 and getting 30, then since it is a right triangle, you can divide that by two and get 15.

You can find D by multiplying 10 by 10 (because it is a square) then you'll get 100 from that. Then you can subtract the rest of the triangles from it which would be 100 minus 30 minus 14 minus 15 and you would get 41 which would be triangle D.

Hope this helps!! Let me know if you need more explanation

Answer:

C

Step-by-step explanation:

[tex]9\sqrt{x} + 3\sqrt{3x} + \sqrt{9x} \\= \sqrt{x} ( 9 + 3\sqrt{3}+3)\\ = 12\sqrt{x} + 3\sqrt{3x}[/tex]

The proportion that correctly defines θ is BC/PC = DE/PE = θ.

option  C.

The length of the arcs is calculated as follows;

Length of arc = (θ/360) x 2πr

where;

For sector PCB, the length of the arc is given as;

(θ/360) x 2π(PC) = BC

(θ/360) x 2π = BC/PC

θ = BC/PC ------- (1)

Note: 2π radian = 360⁰

For sector PED, the length of the arc is given as;

(θ/360) x 2π(PE) = DE

(θ/360) x 2π = DE/PE

θ = DE/PE ------- (2)

Compare the two equations as follows;

BC/PC = DE/PE = θ

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Answer: y=4x+7

Step-by-step explanation:

-8x+2y=14

Convert to y=kx+b

2y=8x+14

y=4x+7

Answer:

The correct option is (D) one of the non-fiction books on the bottom shelf and a second non-fiction book from the bottom shelf.

Step-by-step explanation:

What is probability?

Probability is the branch of mathematics that deals with numerical descriptions of how probable an event is to occur or how likely it is that a claim is true.

To find which 2 books describe a pair of dependent events:

A pair of two dependent events are simply those in which the selection of the second item is contingent on the selection of the first item, causing the probability to change.

Because the sample size is lowered when the initial item is taken without replacement, choosing the exact same item reduces the chance.

Now, in regard to the question, this means that for two of the occurrences to be independent, they must be on the same shelf and of the same type of book, so that the second book cannot be selected until the first one is.

Option D, where both books are on the same shelf and are of the same type, is the only option that fulfills this requirement.

Therefore, the correct option is (D) one of the non-fiction books on the bottom shelf and a second non-fiction book from the bottom shelf.

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The passing score cutoff is 39, and 22.22% of the students failed.

Using the generic formula =

Z = score - mean / standard deviation

According to the question, the results of a psychology test had a mean of 57 and a standard deviation of 9, and they were normally distributed.

Anything 2 or more standard deviations below the mean on the test was considered a failing grade,

hence z = 2/9 = 0.222.

Additionally, the falling cut-off score is determined by using the formula: Falling cut-off = mean - 2 x standard deviation,

or 57 - 2 x 9 = 39.

Hence the passing score cutoff is 39, and 22.22% of the students failed.

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answer is 45

400/9 = 44.44...

so with the extra employees you need another manager.

Answer:

40 managers

Step-by-step explanation:

If there is one manager for every 9 engineers, we can assume that each manager is responsible for a group of 9 engineers. To determine the number of managers in the company, we need to divide the total number of engineers by 9.

First, we need to determine the total number of engineers in the company. Since there is one manager for every 9 engineers, the ratio of engineers to managers is 9:1. This means that for every 10 people (9 engineers and 1 manager), there is 1 manager. So we can find the total number of groups of 10 people by dividing the total number of employees by 10:

400 employees / 10 = 40 groups of 10 people

This means that there are 40 groups of 10 employees (1 manager and 9 engineers) in the company. Since there is 1 manager in each group, the total number of managers in the company is equal to the number of groups:

Number of managers = Number of groups = 40

The solution to the proportional equation 5/8 = 8/a is a = 64/5 or a = 12.8.

To solve the proportional equation 5/8 = 8/a, we can cross-multiply.

Cross-multiplying means multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa.

We have:

5 × a = 8 × 8

5a = 64

To isolate the variable a, we divide both sides of the equation by 5:

a = 64/5

We may cross-multiply the proportional equation 5/8 = 8/a to find the solution.

Cross-multiplication is the process of multiplying the denominator of the second fraction by the first fraction's numerator, and vice versa.

We possess

5 × a = 8 × 8 5a = 64

We divide both sides of the equation by 5 to identify the variable a:

a = 64/5.

We may cross-multiply to find the solution to the proportional equation 5/8 = 8/a.

Cross-multiplication is the process of multiplying one fraction's numerator by another's denominator and vice versa.

There are:

5 × a = 8 × 8 5a

= 64

By multiplying both sides of the equation by 5, we may separate the variable a:

a = 64/5.

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The 97% confidence interval for the proportion of college freshmen who would report cheating if they witnessed it in class is (0.0728, 0.1482).

The point estimate is the proportion of students who responded "yes" out of the total sample.

In this case, the point estimate is 19/172, which is approximately 0.1105.

b. Margin of Error:

To calculate the margin of error, we need the standard deviation. Since we don't have the population standard deviation, we'll use the formula for the estimated standard deviation of a proportion:

Margin of Error (ME) = z√[(p(1 - p)) / n]

Where z is the z-score corresponding to the desired confidence level. For a 97% confidence level, the z-score is approximately 1.96, p is the point estimate of the proportion and n is the sample size.

Using the given values, we can calculate the margin of error:

ME = 1.96√[(0.1105(1 - 0.1105)) / 172]= 0.0377

c. Confidence Interval:

The confidence interval is calculated by subtracting and adding the margin of error from the point estimate.

Confidence Interval = Point Estimate ± Margin of Error

Confidence Interval = 0.1105 ± 0.0377

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Answer:

16 and 4

Step-by-step explanation:

To divide 20 in the ratio 4:1, we need to divide the quantity into 5 parts (4 parts for the first ratio and 1 part for the second ratio) and then allocate the parts accordingly.

The total number of parts is 4+1=5. So, each part represents 20/5 = 4.

To find the share of the first ratio, we multiply the first ratio (4) by the number of parts it represents (4). So, the first share is 4*4 = 16.

To find the share of the second ratio, we multiply the second ratio (1) by the number of parts it represents (1). So, the second share is 1*4 = 4.

Therefore, the quantities in the ratio 4:1 that add up to 20 are 16 and 4, respectively.

Answer:

  (B)  sin(60°) = 4/x

Step-by-step explanation:

You want an equation for finding the hypotenuse of a 30°-60°-90° triangle with the middle-length side being 4 units.

The trig relation between the side opposite and angle and the hypotenuse of the right triangle is ...

  Sin = Opposite/Hypotenuse

In this triangle, this relation means ...

  sin(60°) = 4/x . . . . . matches choice (B)

__

Additional comment

The middle-length side is opposite the middle length angle.

This is a "special" right triangle with sides in the ratios 1 : √3 : 2. The length x will be 4(2/√3) = (8/3)√3. It can be useful to remember the side length ratios for this triangle.

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Answer:

The original number was 4.

Step-by-step explanation:

We can construct an equation to model the given situation, using a variable x to represent the original number:

"the quantity of 6 minus a number"

[tex](6 - x)[/tex]

"four times the quantity"

[tex]4(6 - x)[/tex]

"is 8"

[tex]4(6 - x) = 8[/tex]

We can solve for x in this equation.

[tex]4(6 - x) = 8[/tex]

↓ applying the distributive property ... [tex]A(B+C) = AB + AC[/tex]

[tex]24 - 4x = 8[/tex]

↓ adding 4x to both sides

[tex]24 = 8 + 4x[/tex]

↓ subtracting 8 from both sides

[tex]16 = 4x[/tex]

↓ dividing both sides by 4

[tex]4 = x[/tex]

[tex]\boxed{x = 4}[/tex]

So, the original number was 4.

Answer:

y = - 3x + 7

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (4, - 5 ) and (x₂, y₂ ) = (3, - 2 )

m = [tex]\frac{-2-(-5)}{3-4}[/tex] = [tex]\frac{-2+5}{-1}[/tex] = [tex]\frac{3}{-1}[/tex] = - 3 , then

y = - 3x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (3, - 2 )

- 2 = - 3(3) + c = - 9 + c ( add 9 to both sides )

7 = c

y = - 3x + 7 ← equation of line

Answer: Lara

Step-by-step explanation:

convert yards into feet

yards*3= feet

89 yards * 3 = 267 feet

Since 270 is greater than 267 Lara. Lara's distance is greater by 3 feet since 270-267 is 3

Answer:

21

Step-by-step explanation:

Answer:

The square root of 50 is approximately 7.07.

Answer:

7.07106781187...

Step-by-step explanation:

its an irrational number

(a) The inverse variation equation that relates to x and y is  y = -20/x.

(b) The value of y when x = 25 is -0.8 or -4/5

Inverse variation is the relationships between variables that are represented in the form of y = k/x, where x and y are two variables and k is the constant value.

(a) To write the inverse variation equation that relates to x and y, we look for the constant k connecting x and y

From the questiuon,

To remove the proprotionality/Variation sign, we introduce a constant k

Given:

Substitute into equation 1 and solve for k

Hence, the equation is y = -20/x.

(b) To find the value of y when x = 25, we subtitute into the equation

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The possible trigonometric ratios for the angle θ are given as follows:

A. [tex]\sec{\theta} = \frac{3}{\sqrt{5}}[/tex] and [tex]\tan{\theta} = \frac{2}{\sqrt{5}}[/tex]

B. [tex]\cos{\theta} = \pm \frac{\sqrt{5}}{3}[/tex] and [tex]\tan{\theta} = \frac{2}{\sqrt{5}}[/tex]

The sine for the angle θ is given as follows:

sin(θ) = 2/3.

The identity relating the sine and the cosine is given as follows:

sin²(θ) + cos²(θ) = 1.

Hence the cosine of the angle is given as follows:

cos²(θ) = 1 - 4/9

cos²(θ) = 5/9

[tex]\cos{\theta} = \pm \frac{\sqrt{5}}{3}[/tex]

The tangent is given by the division of the sine by the cosine, hence it is calculated as follows:

[tex]\tan{\theta} = \frac{2}{\sqrt{5}}[/tex]

(if the cosine is negative, the tangent is then negative, as the sine is positive).

The secant is the division of one by the cosine, that is, it is the inverse fraction of the cosine, hence:

[tex]\sec{\theta} = \frac{3}{\sqrt{5}}[/tex]

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The option that would create an appropriate strata is given as follows:

Four teachers from each school in the district are chosen at random to complete a survey on the topic.

Samples may be classified as follows:

An appropriate strata is created when stratified sampling is used that is, the population is divided into groups(each school in the district), and then an equal amount(four teachers from each school) is surveyed.

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Answer:

x ≈ 12.7

Step-by-step explanation:

since the triangle is isosceles then the 2 legs are congruent, both 9

using Pythagoras' identity in the right triangle.

the square on the hypotenuse is equal to the sum of the squares on the other 2 sides , that is

x² = 9² + 9² = 81 + 81 = 162 ( take square root of both sides )

x = [tex]\sqrt{162}[/tex] ≈ 12.7 ( to the nearest tenth )

Answer:

Solution is in the attached photo.

Step-by-step explanation:

This question tests on the concept of triangles, as this is an isosceles triangle, the other 2 unknown angles are the same, as well as the sine rule.

Answer:

Step-by-step explanation:

lets solve all the equations and check:

1 ) 8 - x = 11

 -x = 11 - 8

  x = -3 ------------- not this one

2 ) x + 7 = 4

    x = 4 - 7

    x = -3 -------------not this one

3 ) 5 + x = 9

    x = 9 - 5

    x = 4 ------------- not this one

4 ) x - 2 = 1

    x = 1 + 2

    x = 3 ----------- this is the correct option

                hope this helps!

The perimeter of the polygon which is a triangle is

72 units

The perimeter of the polygon is solved using the knowledge of tangents

This is applied as follows

perimeter of the triangle = GH + HL + GJ

GH = (4 + 32 - 15) = 21

HL = 15 + 4 = 19

GJ = 32

plugging in the values we have

perimeter of the triangle = 21 + 19 + 32

perimeter of the triangle = 72 units

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