In a large school, it was found that 71% of students are taking a math class, 77% of student are taking an English class, and 58% of students are taking both. Find the probability that a randomly selected student is taking a math class or an English class. Write your answer as a decimal, and round to 2 decimal places if necessary.

Answer:

  linear

Step-by-step explanation:

The x-values all differ by 1, which is to say they are equally-spaced. The corresponding y-values all differ by -3. When (first) differences of equally-spaced values of y are constant, the function is of first degree, which is to say it is linear.

___

If second differences are non-zero and constant, the function is of second degree, quadratic.

Answer:

line

Step-by-step explanation:

The graphing option sounds nice...

But  lines have the same slope no matter what two points you choose.

You can see that x is going up by the same number (plus 1) each time and the y's are going down by the same number each time (minus 3) so this says no matter what two points you choose you will have the same slope which means it is a line.

ANSWER

36,-36

EXPLANATION

The given function is:

[tex]f(x) = {x}^{2} + 12x + 6[/tex]

To write this function in vertex form;

We need to add and subtract the square of half the coefficient of x.

The coefficient of x is 12.

Half of it is 6.

The square of 6 is 36.

Therefore we add and subtract 36.

Hence the zero pair is:

36, -36.

The correct answer is D.

Answer:

Last option: 36,-36​

Step-by-step explanation:

The vertex form of the function of a parabola is:

[tex]y=a(x-h)^2+k[/tex]

Where (h,k) is the vertex.

To write the given function in vertex form, we need to Complete the square.

Given the Standard form:

[tex]y=ax^2+bx+c[/tex]

We need to add and subtract [tex](\frac{b}{2})^2[/tex] on one side in order to complete the square.

Then, given [tex]y=x^2+12x+6[/tex], we know that:

[tex](\frac{12}{2})^2=6^2=36[/tex]

Then, completing the square, we get:

 [tex]y=x^2+12x+(36)+6-(36)[/tex]

[tex]y=(x+6)^2-30[/tex] (Vertex form)

Therefore, the answer is: 36,-36​

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

let's recall that the vertex is half-way between the focus point and the directrix, at "p" units away from both.

Let's notice that the focus point is below the directrix, that means the parabola is vertical, namely the squared variable is the "x", and it also means that it's opening downwards as  you see in the picture, namely that "p" is negative, in this case "p" is 1 unit, and thus is -1.

[tex]\bf \textit{parabola vertex form with focus point distance} \\\\ \begin{array}{llll} 4p(x- h)=(y- k)^2 \\\\ \stackrel{\textit{we'll use this one}}{4p(y- k)=(x- h)^2} \end{array} \qquad \begin{array}{llll} vertex\ ( h, k)\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix} \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \begin{cases} h=-2\\ k=5\\ p=-1 \end{cases}\implies 4(-1)(y-5)=[x-(-2)]^2\implies -4(y-5)=(x+2)^2 \\\\\\ y-5=-\cfrac{1}{4}(x+2)^2\implies y=-\cfrac{1}{4}(x+2)^2+5[/tex]

In the [tex]x[/tex] direction we consider the [tex]m=2[/tex] subintervals [0, 1] and [1, 2] (each with length 1), while in the [tex]y[/tex] direction we consider the [tex]n=2[/tex] subintervals [0, 2] and [2, 4] (with length 2). Then the lower right corners of the cells in the partition of [tex]R[/tex] are (1, 0), (2, 0), (1, 2), (2, 2).

Let [tex]f(x,y)=7x+5y^2[/tex]. The volume of the solid is approximately

[tex]\displaystyle\iint_Rf(x,y)\,\mathrm dx\,\mathrm dy\approx f(1,0)\cdot1\cdot2+f(2,0)\cdot1\cdot2+f(1,2)\cdot1\cdot2+f(2,2)\cdot1\cdot2=\boxed{164}[/tex]

###

More generally, the lower-right-corner Riemann sum over [tex]m=\mu[/tex] and [tex]n=\nu[/tex] subintervals would be

[tex]\displaystyle\sum_{m=1}^\mu\sum_{n=1}^\nu\left(7\frac{2m}\mu+5\left(\frac{4n-4}\nu\right)^2\right)\frac{2-0}\mu\frac{4-0}\nu=\frac83\left(101+\frac{21}\mu+\frac{40}{\nu^2}-\frac{120}\nu\right)[/tex]

Then taking the limits as [tex]\mu\to\infty[/tex] and [tex]\nu\to\infty[/tex] leaves us with an exact volume of [tex]\dfrac{808}3[/tex].

The Riemann sum is used to provide an estimate of the volume of a solid under the function surface z = 7x + 5y² and above the rectangle R = [0, 2] × [0, 4]. The rectangle is divided into four equal parts, and the function's value at specific points, multiplied by the area of the base, provides the estimate.

This Mathematics problem requires estimating the volume of a solid under the surface z = 7x + 5y² and above the rectangle R = [0, 2]⨯[0, 4] using a Riemann sum. This method is commonly used in calculus to approximate the definite integral of a function.

Applying a Riemann sum with m=n=2 implies the rectangle is split into 4 equal rectangles for the estimation. Our sample delta x and delta y = rectangle's length/2, for this example, Δx = 2/2 = 1 and Δy = 4/2 = 2. The lower right corners points will be (1,2), (1,4), (2,2) and (2,4).

We then find volume estimates by taking the function's value at these sample points and multiplying it by the area of the base. This gives us: ((7*1 + 5*2²) + (7*1 + 5*4²) + (7*2 + 5*2²) + (7*2 + 5*4²))* (Δx*Δy). Simplifying the expression gives us the estimated volume using the Riemann sum.

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Answer:

[tex]\large\boxed{x=\dfrac{k-14}{5}}[/tex]

Step-by-step explanation:

[tex]5x=k-14\qquad\text{divide both sides by 5}\\\\\dfrac{\not5x}{\not5}=\dfrac{k-14}{5}\\\\x=\dfrac{k-14}{5}[/tex]

Answer:

[tex]5x = k - 14 \\ \frac{5x}{5} = \frac{k - 14}{5} \\ x = \frac{k - 14}{5} [/tex]

Answer:

[tex]\bullet\ \ 0.\overline{15}[/tex]

Step-by-step explanation:

  5/33 = (5·3)/(33·3) = 15/99 = 0.151515151515...

_____

You may recall that 1/9 = 0.11111...(repeating indefinitely). That is, a multiple of 1/9 is a single-digit repeating decimal.

Likewise, 1/99 = 0.01010101...(repeating indefinitely). This means when a 2-digit numerator has 99 as the denominator, the decimal equivalent is that number repeated indefinitely. Any fraction with 999 as the denominator is a 3-digit repeat in decimal; 9999 as the denominator gives a 4-digit repeat, and so on.

Step-by-step answer:

Given:

mean, mu = 200 m

standard deviation, sigma = 30 m

sample size, N = 5

Maximum deviation for no damage, D = 100 m

Solution:

Z-score for maximum deviation

= (D-mu)/sigma

= (100-200)/30

= -10/3

From normal distribution tables, the probability of right tail with

Z= - 10/3

is 0.9995709, which represents the probability that the parachute will open at 100m or more.

Thus, by the multiplication rule, the probability that all five parachutes will ALL open at 100m or more is the product of the individual probabilities, i.e.

P(all five safe) = 0.9995709^5 = 0.9978565

So there is an approximately 1-0.9978565 = 0.214% probability that at least one of the five parachutes will open below 100m

The probability that at least one out of five parachutes causes equipment damage, given that the parachute opening altitude is normally distributed with a mean of 200m and standard deviation of 30m, is approximately 0.2%.

The situation described is a question of probability related to the normal distribution. In this case, we are asked to find the probability of a parachute opening at less than 100m, which will cause damage. First, we need to standardize the value to a z-score. The z-score is calculated by subtracting the mean from the value of interest and dividing by the standard deviation. In this case, it will be (100-200)/30, which equals to about -3.33.

By looking at a z-table or using a statistical calculator we find that the probability of single parachute causing damage is approximately 0.0004. However, the question is interested in the probability of at least one out of five parachutes causing damage. This can be approached as 1 minus the probability of none of the five causing damage, which will be 1 - (1-0.0004)^5. Thus, the resulting probability of equipment damage to the payload of at least one of five independently dropped parachutes is approximately 0.002 or 0.2%.

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Answer:

27 SUVs

Step-by-step explanation:

Let number of ordinary cars be x and SUVs be y

We can write 2 equations and use substitution to solve for the number of SUVs.

"The number of ordinary cars is larger than the number of sport utility vehicles by 59.3%"-

This means that 1.593 times more is ordinary cars (x) than SUVs (y), so we can write:

x  = 1.593y

"The difference between the number of cars and the number of SUVs is 16" -

Since we know ordinary cars are "more", we can say x - y = 16

We can now plug in 1.593 y into x of the 2nd equation and solve for y:

x - y = 16

1.593y - y = 16

0.593y = 16

y = 27 (rounded)

Hence, there are 27 SUVs

Answer:

A. -1.6; not significant

Step-by-step explanation:

The z-score of a data set that is normally distributed with a mean of [tex]\bar x[/tex] and a standard deviation of [tex]\sigma[/tex], is given by:

[tex]z=\frac{x-\bar x}{\sigma}[/tex].

From the question, the test score is: [tex]x=48.4[/tex], the mean is [tex]\bar x=66[/tex], and the standard deviation is [tex]\sigma =11[/tex].

We just have to plug these values into the above formula to obtain:

[tex]z=\frac{48.4-66}{11}[/tex].

This simplifies to:  [tex]z=\frac{-17.6}{11}[/tex].

[tex]z=-1.6[/tex].

We can see that the z-score falls within two standard deviations of the mean.

Since [tex]-2\le-1.6\le2[/tex] the value is not significant.

The correct answer is A. -1.6; not significant

Answer:

Step-by-step explanation:

Given

Find

Solution

a. Solve x plus y equals 81 for y.

  y equals 81 minus x

__

b. Substitute the result from part a into the equation P equals x squared y for the variable that is to be maximized.

  P equals x squared left parenthesis 81 minus x right parenthesis

__

c. Find the domain of the function P found in part b.

  left bracket 0 comma 81 right bracket

__

d. Find dP/dx. Solve the equation dP/dx = 0.

  P = 81x² -x³

  dP/dx = 162x -3x² = 3x(54 -x) = 0

The zero product rule tells us the solutions to this equation are x=0 and x=54, the values of x that make the factors be zero. x=0 is an extraneous solution for this problem so ...

  P is maximized at (x, y) = (54, 27).

The problem is about using the equations x + y = 81 and P=x^2y to find the maximum possible value of P. This involves solving for y, substituting the result into the P equation, determining the domain of P, and finding the derivative of P to solve for x.

We are given the system of equations: x + y = 81 and P = x^2*y and we're tasked with finding the optimum value for P given these constraints.

a. We solve for y in the equation x + y = 81, so y = 81 - x

b. We substitute the result from part a into the equation for P, P = x^2(81 - x)

c. The domain of function P is the set of all possible x values or [0, 81]. This is because x and y are non-negative and y is equal to 81 minus x, meaning the highest x can be is 81

d. With respect to maximization, step d usually involves calculating the derivative of P with respect to x, setting it equal to zero, and solving for x. If you apply the product rule and the chain rule, you would get dP/dx = 0 then solve for x

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Answer:

1.406×[tex]10^{[tex]10^{18}km cubed

Step-by-step explanation:

The volume of a sphere is

[tex]V=\frac{4}{3}\pi r^3[/tex]

Filling in our formula:

[tex]V=\frac{4}{3}\pi (695,000)^3[/tex]

Cubing first gives us:

[tex]V=\frac{4}{3}\pi (3.35702[/tex]×[tex]10^{17}[/tex]

Do the multiplication and division of those numbers, multiply in the value of pi on your calculator, and you'll get 1.406×[tex]10^{18}[/tex]

To determine the volume of the Sun with a radius of about 695,000 km, we first convert the radius to centimeters and then apply the formula V = (4/3)πr³. After performing the calculations, the volume of the Sun is approximately 1.401 x 10³³ cm³in scientific notation with three decimal places in the mantissa.

The student has asked what the volume of the Sun is, given its radius of about 695,000 km. To find the volume of a sphere, the formula to use is V = (4/3)πr³, where V represents the volume and r is the radius.

First, we need to convert the radius from kilometers to centimeters because the standard unit for volume in scientific notation often involves cubic centimeters. There are 100,000 centimeters in a kilometer, so the radius in centimeters is 695,000 km × 100,000 cm/km = 6.95 x 10¹⁰cm.

Now, we can calculate the volume using the formula:

V = (4/3)π(6.95 x 10¹⁰ cm)³

V = (4/3)π(6.95^3 x 10³⁰) cm³

V = (4/3)π(334.14 x 10³⁰) cm³

V = (4/3)π(3.3414 x 10³²) cm³

V ≈ 4.1888 x 3.3414 x 10³² cm³

V ≈ 1.401 x 10^33 cm³

Therefore, the volume of the Sun in scientific notation, using three decimal places in the mantissa, is approximately 1.401 x 10³³ cm³.

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Answer: [tex]\dfrac{1}{6}[/tex]

Step-by-step explanation:

The given event : A randomly drawn marble is blue.

The number of blue marbles in the bag = 2

The total number of marbles in the bag = [tex]2+7+3=12[/tex]

Now, the probability of drawing a blue marble is given by  :-

[tex]\text{P(Blue)}=\dfrac{\text{Number of blue marbles}}{\text{Total number of marbles}}\\\\\Rightarrow\text{P(Blue)}=\dfrac{2}{12}=\dfrac{1}{6}[/tex]

Hence, the probability of the given event event = [tex]\dfrac{1}{6}[/tex]

Answer:

Width = 3 feet

Length = 7 feet

Step-by-step explanation:

x represents the width of doghouse so,

Width = x

Length is 4 times more than its width

Length = x+4

Area of doghouse = 21 square feet

We know

Area of rectangle = Length * Width

21 = (x+4)*x

21 = x^2+4x

=> x^2+4x-21=0

Solving the above quadratic equation by factorization to find the value of x

x^2+7x-3x-21=0

x(x+7)-3(x+7)=0

(x+7)(x-3)=0

x+7 =0 and x-3=0

x= -7 and x =3

Since the width of rectangle can never be negative so, x=3

Width =x = 3 feet

Length = x+4 = 3+4 = 7 feet

Answer:  (x + 7)(x - 3) = 0

               width (x) = 3

               length (x+4) = 7

Step-by-step explanation:

Area (A) = length (l) × width (x)

       21   =  (x + 4)     ×    (x)

       21   =  x² + 4x                   distributed (x) into (x + 4)

        0   =  x² + 4x - 21             subtracted 21 from both sides

        0  = (x + 7)(x - 3)               factored quadratic equation

 0 = x + 7   or    0 = x - 3          applied Zero Product Property

 x = -7        or     x = 3               solved each equation

x = -7 is not valid because lengths cannot be negative

so x = 3

and length ... x + 4 = (3) + 4 = 7

[tex]|\Omega|=52\cdot51\cdot50=132600[/tex]

a)

[tex]|A|=4\cdot3\cdot2=24\\P(A)=\dfrac{24}{132600}=\dfrac{1}{5525}[/tex]

b)

[tex]|A|=13\cdot12\cdot11=1716\\P(A)=\dfrac{1716}{132600}=\dfrac{11}{850}[/tex]

a. Probability of all 3 cards being queens:

b. Probability of all 3 cards being spades:

Answer:

First brand of antifreeze: 21 gallons

Second brand of antifreeze: 9 gallons

Step-by-step explanation:

Let's call A the amount of  first brand of antifreeze. 20% pure antifreeze

Let's call B the amount of second brand of antifreeze. 70% pure antifreeze

The resulting mixture should have 35% pure antifreeze, and 30 gallons.

Then we know that the total amount of mixture will be:

[tex]A + B = 30[/tex]

Then the total amount of pure antifreeze in the mixture will be:

[tex]0.2A + 0.7B = 0.35 * 30[/tex]

[tex]0.2A + 0.7B = 10.5[/tex]

Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -0.7 and add it to the second equation:

[tex]-0.7A -0.7B = -0.7*30[/tex]

[tex]-0.7A -0.7B = -21[/tex]

[tex]-0.7A -0.7B = -21[/tex]

               +

[tex]0.2A + 0.7B = 10.5[/tex]

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

[tex]-0.5A = -10.5[/tex]

[tex]A = \frac{-10.5}{-0.5}[/tex]

[tex]A = 21\ gallons[/tex]

We substitute the value of A into one of the two equations and solve for B.

[tex]21 + B = 30[/tex]

[tex]B = 9\ gallons[/tex]

Answer with explanation:

Given the function f from R  to [tex](0,\infty)[/tex]

f: [tex]R\rightarrow(0,\infty)[/tex]

[tex]-f(x)=x^2[/tex]

To prove that  the function is objective from R to  [tex](0,\infty)[/tex]

Proof:

[tex]f:(0,\infty )\rightarrow(0,\infty)[/tex]

When we prove the function is bijective then we proves that function is one-one and onto.

First we prove that function is one-one

Let [tex]f(x_1)=f(x_2)[/tex]

[tex](x_1)^2=(x_2)^2[/tex]

Cancel power on both side then we get

[tex]x_1=x_2[/tex]

Hence, the function is one-one on domain [tex[(0,\infty)[/tex].

Now , we prove that function is onto function.

Let - f(x)=y

Then we get [tex]y=x^2[/tex]

[tex]x=\sqrt y[/tex]

The value of y is taken from [tex](0,\infty)[/tex]

Therefore, we can find pre image  for every value of y.

Hence, the function is onto function on domain [tex](0,\infty)[/tex]

Therefore, the given [tex]f:R\rightarrow(0.\infty)[/tex] is bijective function on [tex](0,\infty)[/tex] not on  whole domain  R .

Hence, proved.

Final answer:

To determine the probability that the first ball has a smaller number than 4, we count the three balls with numbers 1, 2, and 3, resulting in a 3 out of 11 probability. The probability that the first ball was even is calculated by counting the even-numbered balls (2, 6, 8, 10, 12), which gives a probability of 5 out of 11.

Explanation:

The question asks about the probability that the first ball drawn from a box of 12 numbered balls is smaller or even, given that the second ball drawn has the number 4 on it and the drawing occurs without replacement. We know that once the second ball has been confirmed as the number 4, there are 11 remaining possibilities for the first ball.

To find the probability that the first ball had a smaller number than 4, we count the balls that are numbered less than 4. There are 3 such balls: 1, 2, and 3. Therefore, the probability is 3 out of 11 that the first ball had a smaller number.

To calculate the probability that the first ball had an even number, we consider only the even-numbered balls among the 11 remaining. These are 2, 6, 8, 10, and 12. So, there are 5 even-numbered balls, making the probability 5 out of 11 that the first ball was even.

Answer:

The distance is  4.726

Step-by-step explanation:

we need to find the distance from the point to the line

Given:- point (-1,-2,1) and line ; x=4+4t, y=3+t, z=6-t .

used formula [tex]d=\frac{|a\times b|}{|a|}[/tex]

Let point P be (-1,-2,1)

using value t=0 and t=1

The point Q (4 , 3, 6) and R ( 8, 4, 5)

Let a be the vector from Q to R :   a = < 8 - 4, 4 - 3, 5 - 6 > = < 4, 1, -1 >

Let b be the vector from Q to P:    b = < -1 - 4, -2 - 3, 1 - 6> = < -5, -5, -5 >

The cross product of a and b is:

[tex]a \times b= \begin{vmatrix} i & j & k\\ 4 &1&-1\\-5 &-5&-5\\ \end{vmatrix}[/tex]

= -6i+15j-15k

The distance is : [tex]d=\frac{\sqrt{(-6)^{2}+(15)^{2}+(-15)^{2}}}{\sqrt{(4)^{2}+(1)^{2}+(-1)^{2}}}[/tex]

[tex]=\frac{\sqrt{36+225+225}}{\sqrt{16+1+1}}[/tex]

[tex]=\frac{\sqrt{36+225+225}}{\sqrt{16+1+1}}[/tex]

[tex]d=\frac{\sqrt{486}}{\sqrt{18}}[/tex]

≈4.726

Therefore, the distance is  4.726

Answer:yes

Step-by-step explanation:

Theoretically he could just open the doors leading to the seven cells to the opposing side of the bored, then move forward the seven more, ending in the corner cell diagonal from his orgiu position.

No, I don’t think so. If he passes through each column of cells, he gets to the end cell to the left or right from him instead of the one exactly diagonal from him.

Hope this helps!

Answer:

[tex]\large\boxed{7.\ B.\ 8s^2+16s+1}\\\\\boxed{8.\ D.\ -15t^9u^{11}}\\\\\boxed{11.\ C.\ 3,\ -2}[/tex]

Step-by-step explanation:

[tex]7.\\(3s^2+7s+2)+(5s^2+9s-1)=3s^2+7s+2+5s^2+9s-1\\\\\text{combine like terms}\\\\=(3s^2+5s^2)+(7s+9s)+(2-1)\\\\=8s^2+16s+1[/tex]

[tex]8.\\(-3t^2u^3)(5t^7u^8)=(-3\cdot5)(t^2t^7)(u^3u^8)\qquad\text{use}\ a^na^m=a^{n+m}\\\\=-15t^{2+7}u^{3+8}=-15t^9u^{11}[/tex]

[tex]11.\\n-the\ number\\\\n^2=n+6\qquad\text{subtract}\ n\ \text{and}\ 6\ \text{from both sides}\\\\n^2-n-6=0\\\\n^2+2n-3n-6=0\\\\n(n+2)-3(n+2)=0\\\\(n+2)(n-3)=0\iff n+2=0\ \vee\ n-3=0\\\\n+2=0\qquad\text{subtract 2 from both sides}\\n=-2\\\\n-3=0\qquad\text{add 3 to both sides}\\n=3[/tex]

By Stokes' theorem,

[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S[/tex]

where [tex]S[/tex] is the surface with [tex]C[/tex] as its boundary. The curl is

[tex]\nabla\times\vec F(x,y,z)=2\,\vec\imath-x\,\vec k[/tex]

Parameterize [tex]S[/tex] by

[tex]\vec\sigma(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(8-u\cos v)\,\vec k[/tex]

with [tex]0\le u\le9[/tex] and [tex]0\le v\le2\pi[/tex]. Then take the normal vector to [tex]S[/tex] to be

[tex]\vec\sigma_u\times\vec\sigma_v=u\,\vec\imath+u\,\vec k[/tex]

Then the line integral is equal to the surface integral,

[tex]\displaystyle\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^9(2\,\vec\imath-u\cos v\,\vec k)\cdot(u\,\vec\imath+u\,\vec k)\,\mathrm du\,\mathrm dv[/tex]

[tex]\displaystyle=\int_0^{2\pi}\int_0^9(2u-u^2\cos v)\,\mathrm du\,\mathrm dv=\boxed{162\pi}[/tex]

Final answer:

To find the total number of different value meal packages possible at Ron's Subs, multiply the number of choices for drinks (4), sandwiches (3), and chips (3), resulting in 4 × 3 × 3 = 36 possible combinations.

Explanation:

To find the total number of different value meal packages possible, we calculate the product of the number of choices for each category. In this case, there are 4 types of drinks, 3 types of sandwiches, and 3 types of chips. Thus, the calculation is as follows:

To find the total number of combinations, we multiply the number of choices for each category:

4 (drinks) × 3 (sandwiches) × 3 (chips) = 36 different value meal packages.

Therefore, at Ron's Subs, there are 36 possible different value meal packages a customer can choose from.

Answer:

Type of function: Absolute Value

Transformations: 1) elongated by a stretch factor of 2; 2) shifted left 3; 3) shifted down 5

Answer:

Step-by-step explanation:

The parent function is

[tex]f(x)=|x|[/tex]

The transformed function is

[tex]g(x)=2|x+3|-5[/tex]

You can deduct by comparison, that the function was shifted 5 units down, 3 units to the left, and vertically streched by a scale of 2.

We deduct this transfromations based on the following rules.

[tex]f(x)-u[/tex] indicates a movement downside [tex]u[/tex] units.

[tex]f(x+u)[/tex] indicates a movement leftside [tex]u[/tex] units.

[tex]uf(x)[/tex] indicates a vertical stretch for [tex]u>1[/tex].

Answer:

The range is 21

The interquartile range is 6

The variance is 33

The standard deviation (σ) is 5.74

Step-by-step explanation:

* Lets study the information to solve the problem

- The values of the data are 27 , 24 , 23 , 15 , 30 , 36 , 29 , 24

- They are eight values

* lets arrange them from small to big

∴ The values are 15 , 23 , 24 , 24 , 27 , 29 , 30 , 36

* Now lets solve the problem

# The range

- It is the difference between the largest and the smallest values

∵ The largest value is 36

∵ The smallest value is 15

∴ The range = 36 - 15 = 21

* The range is 21

# The interquartile range

- The steps to find the interquartile range is:

1- Arrange the values from the smallest to the largest

∴ The values are 15 , 23 , 24 , 24 , 27 , 29 , 30 , 36

2- Find the median

- The median is the middle value after arrange them

* If there are two values in the middle take their average

∵ The values are 8 then the 4th and the 5th are the values

∵ The 4th is 24 and the 5th is 27

∴ The median = [tex]\frac{24+27}{2}=\frac{51}{2}=25.5[/tex]

∴ The median is 25.5

3- Calculate the median of the lower quartile

- The lower quartile is the median of the first half data values

∵ There are 8 values

∴ The first half is the first four values

∴ The first half values are 15 , 23 , 24 , 24

∵ The middle values are 23 and 24

∴ The median of lower quartile = [tex]\frac{23+24}{2}=\frac{47}{2}= 23.5[/tex]

- Similar find the median of the upper quartile

- The upper quartile is the median of the second half data values

∵ There are 8 numbers

∴ The second half is the last four values

∴ The second half values are 27 , 29 , 30 , 36

∵ The middle values are 29 and 30

∴ The median of upper quartile = [tex]\frac{29+30}{2}=\frac{59}{2}=29.5[/tex]

4- The interquartile range (IQR) is the difference between the upper

    and the lower medians

∴ The interquartile range = 29.5 - 23.5 = 6

* The interquartile range is 6

# The variance

- The variance is the measure of how much values in a set of data are

  likely to differ from the mean value of the same data

- The steps to find the variance

1- Find the mean of the data

∵ The mean = sum of the data ÷ the number of the values

∵ The sum = 15 + 23 + 24 + 24 + 27 + 29 + 30 + 36 = 208

∵ The number of values is 8

∴ The mean = [tex]\frac{208}{8}=26[/tex]

∴ The mean is 26

2- Subtract the mean from each value and square the answer

∴ 15 - 26 = -11 ⇒ (-11)² = 121

∴ 23 - 26 = -3 ⇒ (-3)² = 9

∴ 24 - 26 = -2 ⇒ (-2)² = 4

∴ 24 - 26 = -2 ⇒ (-2)² = 4

∴ 27 - 26 = 1 ⇒ (1)² = 1

∴ 29 - 26 = 3 ⇒ (3)² = 9

∴ 30 - 26 = 4 ⇒ (4)² = 16

∴ 36 - 26 = 10 ⇒ (10)² = 100

3- Add all of these squared answer and divide the sum by the number

   of the values

∴ The sum = 121 + 9 + 4 + 4 + 1 + 9 + 16 + 100 = 264

∵ They are 8 values

∴ The variance (σ²) = [tex]\frac{264}{8}=33[/tex]

* The variance is 33

# The standard deviation

- It is the square root of the variance

∵ The variance = 33

∴ The standard deviation (σ) = √33 = 5.74

* The standard deviation (σ) is 5.74

Answer: Probability that the entire batch will be rejected is 0.611.

Step-by-step explanation:

Since we have given that

Number of clock radios in a batch = 8000

Probability of defective clock radio = 7%

According to question, we have mentioned that A sample of 13 clock radios is randomly selected without replacement from the​ 8,000 and tested.

We will use "Binomial distribution":

here, n = 13 and

p (probability of success) = 7% = 0.07

so, we need to find that

P(the entire batch will be rejected)  = P(at least one of those test is defected)

So, it becomes,

P(at least one of those tested is defective) = 1 - P(none are defective)

So, P(none are defective ) is given by

[tex](1-0.07)^{13}\\\\=0.93^{13}\\\\=0.389[/tex]

So, P(at least one of those tested is defective) = 1 - P(none are defective)

                                                                             =  1 -  0.389

                                                                             =  0.611

Hence, Probability that the entire batch will be rejected is 0.611.

Answer:

x = -6 and y = 9

Step-by-step explanation:

It is given that,

0.12x - 0.07y = -1.35    -----(1)

0.4x + 0.8y = 4.8   -----(2)

To find the value of x and y

eq(1) * 100 ⇒

12x + 7y = -135   -----(3)

eq(2) /0.4 ⇒

x + 2y = 12   -----(4)

eq(4) * 12 ⇒

12x + 24y = 144   ---(5)

eq(5) - eq(3) ⇒

12x + 24y = 144   ---(5)

12x - 7y = -135  -----(3)

 0  + 31y = 279

y = 279/31 = 9

Substitute the value of y in eq(4)

x + 2y = 12   -----(4)

x + 2*9 = 12

x = 12 - 18 = -6

Therefore x = -6 and y = 9

The mean value (expected value) of the given discrete variable x is,

⇒ 2.35.

The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.

Now, For the mean value of the given discrete variable x, we need to multiply each value of x by its corresponding probability, and then add up these products.

Hence, the expected value of x can be calculated as follows:

Mean = (0 x 0.2) + (1 x 0.2) + (2 x 0.15) + (3 x 0.15) + (4 x 0.15) + (5 x 0.1) + (6 x 0.05)

Simplifying this expression, we get:

Mean = 0 + 0.2 + 0.3 + 0.45 + 0.6 + 0.5 + 0.3

Mean = 2.35

Therefore, the mean value (expected value) of the given discrete variable x is 2.35.

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Answer:

12m

Step-by-step explanation:

We are given the following expression where a fraction is divided by another fraction:

[tex]\frac{\frac{16m^2}{m^2+5} }{\frac{4m}{3m^2+15} }[/tex]

To change this division into multiplication, we will take reciprocal of the fraction in the denominator and then solve:

[tex] \frac { 1 6 m ^ 2 } { m^2+5} } \times \frac{3m^2+15}{4m}[/tex]

Factorizing the terms to simplify:

[tex] \frac { 4 m ( 4m ) } { m ^ 2 + 5 } \times \frac { 3 ( m ^ 2 + 5 ) } { 4 m } [/tex]

Cancelling the like terms to get:

12m

Answer: [tex]12m[/tex]

Step-by-step explanation:

Given the expression [tex]\frac{\frac{16m^2}{m+5}}{\frac{4m}{3m^2+15}}[/tex], we can rewrite it in this form:

[tex](\frac{16m^2}{m+5})(\frac{3m^2+15}{4m})[/tex]

Now we must multiply the numerator of the first fraction by the numerator of the second fraction and   the denominator of the first fraction by the denominator of the second fraction:

 [tex]=\frac{(16m^2)(3m^2+15)}{(m^2+5)(4m)}}[/tex]

According to the Quotient of powers property:

[tex]\frac{a^m}{a^n}=a^{(m-n)}[/tex]

And the Product of powers property states that:

[tex](a^m)(a^n)=a^{(m+n)}[/tex]

Then, simplifying, we get:

[tex]=\frac{3(m^2+5)(4m)(4m)}{(m^2+5)(4m)}}\\\\=3(4m)\\\\=12m[/tex]

Answer:

the smallest angle is x, and then the second angle is 3x and the third angle is 3x+50. I don't know what the smallest angle is though.

Step-by-step explanation:

first I don't see the smallest angle that's why I called it x. Next you said 3 times the smallest angle so 3 times x. Finally you said the third angle is 50 more than the second one so you add 50 to 3x. I can't tell you what the answer is though because you didn't tell me the smallest angles measure.

Answer:

[tex]x_2 \approx -1.769[/tex]

Step-by-step explanation:

Let [tex]f(x)=x^3+x+7[/tex]

So [tex]f'(x)=3x^2+1[/tex]

[tex]x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}[/tex]

Let [tex]x_1=-2[/tex]

We are going to find [tex]x_2[/tex]

So we are evaluating [tex]-2-\frac{f(-2)}{f'(-2)}[/tex]

First step find f(-2)

Second step find f'(-2)

Third step plug in those values and apply PEMDAS!

[tex]f(-2)=(-2)^3+(-2)+7=-8-2+7=-10+7=-3[/tex]

[tex]f'(-2)=3(-2)^2+1=3(4)+1=12+1=13[/tex]

So

[tex]x_2=-2-\frac{-3}{13} \\\\ x_2=\frac{-26+3}{13} \\\\ x_2=\frac{-23}{13} \\\\ x_2 \approx -1.769[/tex]

The second approximation x2 using Newton's method for the equation x3 + x + 7 = 0 with an initial approximation of x1 = -2 is -2.2764.

In order to find the second approximation x2 using

Newton's method

, we need to use the definition of Newton's method, which states that: x

n+1

= x

n

- f(x

n

)/f'(x

n

). Here, our function f(x) is x

3

+ x + 7. The derivative, f'(x), is 3x

2

+ 1. If our initial approximation, x1, is -2, we can substitute these values into our method to find x2. So, x2 = x1 - f(x1)/f'(x1) = -2 - ((-2)^3 + (-2) + 7) / (3*(-2)^2 + 1) = -2.2764 (rounded to four decimal places).

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