Let f(x) = -3x² - 1 and g(x) = -5x. Find f(g(x)).
-15x²-1
0-75x²-1
15x²+5
15x² + 5x

The perimeter of 2nd right isosceles triangle is 16(√2 + 2).

If a is the equal sides of a right isosceles triangle,

Then its hypotenuse is √2 a and perimeter is √2a (√2  + 1 )

The perimeter of 1st right isosceles triangle is 16(√2 + 1)

= √2a (√2  + 1 )

= 16 (√2  + 1 )

a = 8√2

The hypotenuse of 1st triangle = √2  a

= √2 * 8√2

= 16cm

Let A be the equal sides of 2nd right isosceles triangle

Then its hypotenuse is √2 a and perimeter is √2a (√2  + 1 )

According to the question,

A = 16 cm

The perimeter of 2nd right isosceles triangle is √2 * 16(√2 + 1)

= 16(√2 + 2)

Hence, The perimeter of 2nd right isosceles triangle is 16(√2 + 2).

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Answer:

16.2

Step-by-step explanation:

use long division so:

Calculations in mechanics involving masses dispersed in space, such as the linear and angular momentum, benefit from using the centre of mass as a reference point.

The centre of mass of the trapezoid can be found by breaking it into two trapezoids and finding the centre of mass of each. The centre of mass of the first trapezoid is located at (3, 5). The centre of mass of the second trapezoid is located at (9, 5). Therefore, the centre of mass of the trapezoid is located at (6, 5).

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The unit rate of rainfall for 72 hours of rain will be equal to u = 8 inches/hour.

The four basic mathematical operations are the addition, subtraction, multiplication, and division of two or even more integers. Among them is the examination of integers, particularly the order of actions, which is crucial for all other mathematical topics, including algebra, data organization, and geometry.

As per the provided data in the question,

Total time, t = 72 hours and,

The total amount of water in inches, w = 9 inches.

Then unit rate, u will be,

u = 72/9

u = 8 inches/hour.

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The number of ways 3 courses can be selected is 3654.

What are the Permutation and Combination?

Permutations and combinations are the various ways in which objects from a set can be selected to form subsets, generally without replacement. When the order of selection is a factor, this selection of subsets is called a permutation; when it is not, it is called a combination.

Total no. of courses available = 29

Courses needed to graduate = 3

No of ways these courses can be selected:

                    = ²⁹C₃

                    = 29!/ [(29-3)! - 3!]

                    = 29! / 26! 3!

                    = 29 * 28 * 27 * 26!/ 26! 3*2

                    = 29* 14* 9

                    = 3654

So, the number of ways 3 courses can be selected is 3654.

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The probability he or she selects a management major is 0.176. This is estimated using empirical probability.

Probability is a branch of mathematics that deals with numerical probabilities of events. It is expressed in a range from 0 to 1, not below or above this range. The probability formula is P(E) = (number of favorable outcomes)÷(total number of outcomes).

a) Here, given the total number of outcomes is 34. The favorable outcome is 6. Therefore, the probability he or she selecting the management major is,

P(E)=6/34=0.176.

b) The empirical probability employs numerous trials to calculate likelihood using real observed frequencies. It is an experimental probability that is based on previous or past data.

In the given question, we used past data to estimate the probability. Therefore, it is empirical.

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Answer: 6 and 7

Step-by-step explanation:

Find the square root of 42 = 6.48

Closest whole numbers to 6.48 are 6 and 7

Answer: y = 40, slope is 0, the line is horizontal

Step-by-step explanation:

The earth is 40000 kilometers around. There is no need to round off because the circumference already contains significant figures.

How do I calculate a circle's circumference?

A circle's diameter is multiplied by to determine its circumference (pi). You can also determine the circumference by multiplying 2radius by pi (=3.14).

Given: theta angle (central) = 7.2°; arc length S = 800 km

This indicates that an 800 km long arc is extending a circle with a 7.2° center angle ( earth in this case).

By definition, "An arc's length is a portion of a circle's diameter."

Therefore, we must first establish what percentage of the circle the specified arc length represents.

360° is a complete circle.

Thus, the fraction equals 7.2°/360°, or 1/50.

As a result, the stated arc length (800km) with a 7.2° central angle is 1/50th of the entire circle.

Thus, the whole circumference is 800 x 50, or 40000 km.

Alternately, you can calculate circumference using the formula below:

360° center angle x arc length as the circumference (theta)

Values substituted: circumference 800 = 360° 7.2°

circumference = 360°, 800°, and 7.2°, or 40000 kilometers

Therefore, the earth is 40000 kilometers around. There is no need to round off because the circumference already contains significant figures.

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A. Using the distance formula, we can state that the opposite sides are congruent because AD = BC = √10 units and AB = CD = √40 units.

B. The diagonals are equal, AC = BD = √50 units.

C. Based on the slopes of each side, there are 4 right angles on the rectangle.

The distance formula is used to find the distance that exist between tow points that are on a coordinate plane. The formula is: d = [tex]\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex].

What is the Slope of a Line?

Slope = change in y / change in x.

A. The coordinates of each of the vertices of the rectangle are:

A(1, 2)

B(7, 4)

C(8, 1)

D(2, -1)

Use the distance formula to find AB, CD, BC, and AD.

AB = √[(7−1)² + (4−2)²]

AB = √40

CD = √[(2−8)² + (−1−1)²]

CD = √40

BC = √[(8−7)² + (1−4)²]

BC = √10

AD = √[(2−1)² + (−1−2)²]

AD = √10

Therefore, the opposite sides are congruent because AD = BC = √10 units and AB = CD = √40 units.

B. The diagonals are AC and BD. Find their lengths using the distance formula:

AC = √(8−1)² + (1−2)²]

AC = √50 units

BD = √[(2−7)² + (−1−4)²]

BD = √50 units

Therefore, the diagonals are equal, AC = BD = √50 units.

C. Find the slope of AB, CD, BC, and AD:

Slope of AB = change in y / change in x = rise/run = 2/6 = 1/3

Slope of CD = 2/6 = 1/3

Slope of BC = -3/1 = -3

Slope of AD = -3/1 = -3

-3 is the negative reciprocal to 1/3, this means that, if the two lines that meet at a corner have these two slope, then they will form a right angle because they are perpendicular to each other.

Therefore, there are 4 right angles on the rectangle.

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a) scatter diagram made using the provided data is attached.

b) According to the scatter diagram, the number of defective parts discovered rises with line speed.

c) Y = 27.5 - 0.3 X is the regression model.

What does statistics mean in plain English? a field of mathematics concerned with the gathering, examination, presentation, and interpretation of vast amounts of numerical data. a compilation of numerical data

20 faulty pieces are anticipated to be present when the line speed is 25 feet per minute.

Detailed explanation:

The given information is displayed as follows:

Line Speed;                   20,   20,    30,    30,    40,    40,    50,     50

Number of Defective;   23,    21,     19,     16,     15,     17,     14,       11  

Parts located

a. The scatter diagram made with Microsoft Excel is attached.

b. The scatter figure appears to show a relationship between increased line speed and the quantity of detected defective parts.

c. The following equation is given for linear regression using the least squares approach;

Y = a + b·X

Where;

b= N∑XY - (∑X)(∑Y)/N∑X^2 - (∑X)^2

a= ∑Y - b∑X / N

We have data from Microsoft Excel;

∑X = 280, ∑Y = 136, ∑X² = 10,800, ∑XY = 4,460, (∑X)² = 78,400, N = 8

By entering the values, we obtain;

b = (8×4,460 - 280×136)/(8×10,800 - 78,400) = -0.3

a = (136 - (-0.3)×280)/8 = 27.5

As a result, we have the following linear regression;

Y = 27.5 - 0.3·X

Consequently, we have when the line speed is 25 feet per minute;

Y = 27.5 - 0.3 × 25 = 20

When the line speed is 25 feet per minute, Y = 20 faulty components should be expected to be identified.

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Check the picture below, so the parabola looks more or less like so.

keeping in mind the vertex is always half-way between the focus point and the directrix, let's also notice the "p" distance.

[tex]\textit{vertical parabola vertex form with focus point distance} \\\\ 4p(y- k)=(x- h)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h,k+p)}\qquad \stackrel{directrix}{y=k-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{"p"~is~negative}{op ens~\cap}\qquad \stackrel{"p"~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\begin{cases} h=-4\\ k=2\\ p=-1 \end{cases}\implies 4(-1)(y-2)=( ~~ x-(-4) ~~ )^2\implies -4(y-2)=(x+4)^2 \\\\\\ y-2=-\cfrac{1}{4}(x+4)^2\implies {\Large \begin{array}{llll} y=-\cfrac{1}{4}(x+4)^2 + 2 \end{array}}[/tex]

Answer:

x = 0

Step-by-step explanation:

1. Distribute

2. Add the numbers

3. Rearrange terms

4. Subtract 19 from both sides

5. Simplify

6. Add 5x to both sides

7. Simplify

8. Divide both sides by the same factor

9. Simplify

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Step-by-step explanation:

The required relationship between weight and number of months is given as y = 6x + 90.

As of the given data, assume the relationship on the graph,
Initially ordered pair (0, 90) and instantaneous ordered pair (8, 138)

The slope of the line is a tangent angle made by line with horizontal. i.e. m =tanx where x in degrees.

Here,
The slope of the relationship,

M = (y₂ - y₁) /  (x₂ - x₁)

m = 138 - 90 / 8  - 0
m = 48 / 6
m = 6

Now,
Equation of the relationship,

y - y₁ = m (x - x₁)
y - 90 = m{x - 0}
y = 6x + 90

Thus, y = 6x + 90 is the needed relationship between weight and the number of months.

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The minimum power input needed to drive this escalator is

15.59 kw.

What is escalator?

A moving staircase that transports people between floors of a building or other structure is called an escalator. It is made up of a motor-driven chain of separately linked steps on a track that rotates on two other tracks, keeping them horizontal.

Given:

[tex]N_p = 50\\m_p = 7.5 kg\\V = 0.6m/s\\[/tex]

θ = 45°

Total mass = 50 x 75 = 3750 kg

The energy required = Change in potential energy

ΔE = mg ΔZ

Power = ΔE/t = mg(ΔZ/t)

ΔZ/t = vsinθ = 0.6 sin(45°) = 0.4247

Power = mg x 0.4243 = 3750 x 9.8 / 100 x 0.4243

P = 15.59 kw

Hence,  the minimum power input needed to drive this escalator is

15.59 kw.

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The fraction of table filled is 285/301 or in decimal as 0.946

Fractions are expressed in mathematics as a number value that defines a part of a whole or group of objects. Fraction is a component or sector of a given quantity taken from the whole, which might be a number, a specified value, or an item.

In a given fraction, the value on the top is called the numerator, while the value on the bottom is known as the denominator. The numerator defines the number of equal parts taken, whereas the denominator defines the total number of equal parts in a whole.

In this question, we have to add the total number of attendees first

212 + 89 = 301

But each seat except 1 is filled by 16 people

Let's subtract 16 from the total number

301 - 16  = 285

The fraction of table filled is  285 ÷ 301 = 285/301 = 0.946

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The key reason why this investigation's findings are erroneous is sampling bias. In statistics, sampling bias refers to a bias during which a sample is constructed in a way that makes some members of the intended population less likely than others to be included.

A private university's 100 students, 50 men, and 50 women were randomly chosen to participate in a study on how American college students felt about commuting. They were questioned about their daily commuting distance (in minutes) to college. The key reason why this investigation's findings are erroneous is sampling bias.

When certain individuals of a population are consistently more likely to be chosen in a sample than others, this is known as sampling bias. In the medical sciences, it is also known as ascertainment bias. Because sampling bias jeopardizes external validity, particularly population validity, it restricts the generalizability of findings.

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Answer:

5

Step-by-step explanation:

 25/5 = 5/1 = 5

The dimension of the rectangular pen that maximizes the area is 61,250ft.

What is a dimension?

Something's size, including its breadth, height, and depth: She took exact measurements of the room's dimensions in one direction, like the distance from the ceiling to the floor. The room was surprisingly small, in terms of length, width, and height.

Here, we have

L+2W = 700

Area = A= LW = (700-2W)W = 700W -2W²

Take the derivative, set it equal to zero, and solve for W

-4W + 700 = 0

W = 175 feet

L = 700-2(175) = 350 feet

350 by 175 ft maximizes area = 61,250 square feet

Hence, the dimension of the rectangular pen that maximizes the area is 61,250ft.

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Answer:

  see attached

Step-by-step explanation:

You want a graph of the piecewise function ...

[tex]f(x)=\begin{cases}x&\text{if $x\le -1$}\\-3x+2&\text{if $-1 < x < 4$}\\4&\text{if $x\ge 4$}\end{cases}[/tex]

The graph is attached.

For x ≤ -1, the graph is a line with slope 1. If it were allowed to continue, it would pass through the origin. It terminates in a closed dot at (-1, -1)

For -1 < x < 4, the graph is a line with slope -3 through point (0, 2) on the y-axis. It has open circles at both ends, at points (-1, 5) and (4, -10).

For x ≥ 4, the line is horizontal at y=4. It has a closed dot at (4, 4).

__

Additional comment

The dot at the end of an interval is closed if the "or equal to" case is included (≤ or ≥). If it is not (< or >), then the dot is an open circle.

The slope is the ratio of "rise" to "run". It is the number of grid squares the line moves up (+) or down (-) for each grid square to the right.

The probability of getting a prime number of heads given that heads occurs on at least 6 of the tosses is 0.0078

The number of observations n is 7

Here the success is heads

The probability of heads is 1/2

We need to find the the probability of getting a prime number of heads.

Heads occurs on at least 6 of the tosses

prime number greater than equal to 6 and less than equal to 7 is 7

So, X = 7

P(X = 7) = 7C7(1/2)⁷(1/2)⁰

P(X = 7) =  1 x (1/2)⁷ x

P(X =7 ) = 1/128

= 0.0078

Therefore, the probability of getting a prime number of heads given that heads occurs on at least 6 of the tosses is 0.0078

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[tex]\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &20000\\ r=rate\to 10\%\to \frac{10}{100}\dotfill &0.1\\ t=years\\ \end{cases} \\\\\\ A=20000(1 - 0.1)^{t} \implies A=20000(0.9)^t[/tex]

how long till it's $7000?

[tex]\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill &7000\\ P=\textit{initial amount}\dotfill &20000\\ r=rate\to 10\%\to \frac{10}{100}\dotfill &0.1\\ t=\textit{elapsed time} \end{cases} \\\\\\ 7000=20000(1 - 0.1)^{t} \implies \cfrac{7000}{20000}=0.9^t\implies \cfrac{7}{20}=0.9^t[/tex]

[tex]\log\left( \cfrac{7}{20} \right)=\log(0.9^t)\implies \log\left( \cfrac{7}{20} \right)=t\log(0.9) \\\\\\ \cfrac{\log\left( \frac{7}{20} \right)}{\log(0.9)}=t\implies 9.96\approx t \qquad \textit{about 9 years and 11 months}[/tex]

When chemist mixes Chemical A and Chemical B to make a solution

Part A

When the quantity of chemical B = 48 ml, then the quantity of chemical A = 4 ml

Part B

When the quantity of chemical A = 9 ml, then the quantity of chemical B = 108 ml

A chemist mixes Chemical A and Chemical B to make a solution

She uses the same ratio of chemicals for each batch

The solution A and B are in linear relationship

y = kx

Where y is the quantity of chemical A

x is the quantity of chemical B

k is the constant

Substitute the values in the table and find k

36 = k × 3

k = 36 /3

k = 12

Part A

When chemical B = 48 ml

48 = 12 × x

x = 48 / 12

x = 4 ml

Part B

When chemical A = 9 ml

y = 12 × 9

y = 108 ml

Therefore, when Chemical B = 48 ml, chemical B is 4ml . When chemical A = 9 ml, Chemical B = 108 ml

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False, the margin of error of a poll, assuming 95% confidence's measure of an estimate's accuracy.

A measure of an estimate's accuracy that also accounts for the variability of the sampling technique used to produce the estimate is the margin of error of a confidence interval. It doesn't directly take into account sample process bias.

Bias is the term for deliberate mistakes made during the planning or execution of a study that produce outcomes that are not accurate representations of the real population under investigation. Although it is not a precise indicator of bias, bias can have an impact on the margin of error if it causes the sampling technique to be more variable.

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The measure of angle WZY of the Trapezoid WXYZ is; 37

We are told that the Trapezoid WXYZ is isosceles. This means that there would be two angles that are equal.

Thus, we can see that Side WZ and XY are equal and as such, it means that Angle WZY is congruent to angle XYZ.

We are told Angle WZY measures (8x – 3) and angle XYZ measures (7x + 2). Thus;

(8x – 3) = (7x + 2)

8x - 7x = 2 + 3

x = 5

Thus;

Angle WZY = (8(5) – 3)

Angle WZY = 37°

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The original number is -21/5 or -4.2 which is divided by 3 more than the number.

In mathematics, an equation is a statement or formula which shows two expressions having equal value and connected by an equal sign. The two expressions consist of variables and/or numbers.

Let the number be 'x' Two times the number = 2x3 more than the number = x + 3

According to the question,two times a number is divided by 3 more than the number is equal to 7 which means 2x / (x+3) = 7

By applying cross multiplication,2x = 7 × ( x+3)2x

= 7x + 7×32x

= 7x + 212x - 7x

= 21-5x = 21x = 21/ (-5)x

= -4.2

Hence the original number is -4.2

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The perimeter of the window to the nearest hundredth will be;

⇒ 102.8 cm

What is Circle?

The circle is a closed two dimensional figure , in which the set of all points is equidistance from the center.

Given that;

The shape of window is semicircle.

And, The radius of window (r) = 20 cm

Now,

We know that;

The perimeter of semicircle = πr + 2r

Here, The radius of window (r) = 20 cm

So, We get;

The perimeter of the window = πr + 2r

                                             = 3.14 x 20 + 2 x 20

                                             = 62.8 + 40

                                             = 102.8 cm

Thus, The perimeter of the window to the nearest hundredth will be;

⇒ 102.8 cm

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Answer:

it is b 5/7

Step-by-step explanation:

bc you rise 5 and run 7

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The correct option B. 9(5 + 8) ; is the equivalent expression to 45 + 72 using the GCF and the distributive property.

A mathematical statement made up of variables, constants, and mathematical operators is called an expression.

The formula is 45 + 72.

45 and 72 divided by one is three and nine.

The GCF is nine.

The largest number that evenly divides two or even more given numbers is known as the greatest common factor.

You can write the phrase as,

9 ( 5 + 8)

45 +72 = 9 ( 5 + 8)

Thus, the Distributive property was used to factor this.

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The correct question is-

Which correctly shows how to use the GCF and the distributive property to find an expression equivalent to 45 + 72?

A. 3(15 + 24)

B. 9(5 + 8)

C. (5)(9) + (2)(36)

D. (3)(15) + (8)(9)

The first set of steps solving 5x-y = 4 to get y = 5x-4 is correct.

The second set of steps are incorrect since it involves circular logic. Notice we're plugging y = 5x-4 back into the original equation we just solved. It would be like plugging an extension cord into itself, and somehow expecting electricity to magically come out of nowhere.

Instead we should plug that y expression into the first equation and solve for x like so:

8x+2y = -12

8x+2(5x-4) = -12

8x+10x-8 = -12

18x-8 = -12

18x = -12+8

18x = -4

x = -4/18

x = -2/9

Then we use this to find the value of y.

y = 5x-4

y = 5(-2/9)-4

y = -10/9-4

y = -10/9 - 36/9

y = (-10-36)/9

y = -46/9

In summary we have the solution (x,y) = ( -2/9, -46/9 )