Graph the linear equation.
42 + 6y = -12
Plot two points on the line to graph the line.

Answer:

j = - 9/14

Step-by-step explanation:

-4(7j+2) = 10

Distributing First

-28j - 8 = 10

Try to get the variable on one side.

-28j = 18

Divide both sides by -28

j = -18/28 = - 9/14

Answer:

To calculate the scale of the map, we can use the following formula:

Scale = Actual distance / Map distance

(a) Fractional scale:

The actual distance between Point C and Point D is 500 miles, and the distance on the map is 5 inches. Therefore, the fractional scale can be calculated as:

Scale = 500 miles / 5 inches

Scale = 100 miles per inch

So the fractional scale of the map is 1 inch = 100 miles.

(b) Written scale:

To express the scale in written form, we can use the ratio of inches to miles. Since 1 inch represents 100 miles, we can write the scale as:

1 inch represents 100 miles

Alternatively, we can simplify the scale to a more common ratio by dividing both sides by 100:

1/100 inch represents 1 mile

Therefore, the written scale of the map is 1/100 inch = 1 mile.

Answer:

Step-by-step explanation:

To find the total length of both pieces of fabric in centimeters, we need to add the length of the first piece of fabric (142 cm) and the length of the second piece of fabric (2 meters).

However, we need to make sure that the units are consistent before we add the lengths. We can convert the length of the second piece of fabric from meters to centimeters by multiplying by 100. Therefore, the total length in centimeters is:

142 cm + 2 meters * 100 cm/meter = 142 cm + 200 cm = 342 cm

The option that correctly gives the answer is "Multiply 2 by 100, then add 142" (Option C).

The amount a student received in merit scholarships was $3,456 ($478 per student). The cost of full tuition was $4,200. This means that the difference between the amount of the scholarship and the cost of tuition was $744.

Amount is a numerical value that represents a quantity of something. It is used to measure the size, amount, or degree of something, often in terms of money, time, or distance. Amounts are usually expressed in a specific unit, such as dollars, minutes, or kilometers. Amounts can also refer to the total number of something, such as the amount of people in a room or the amount of items in a box. Amounts can also be used to describe a portion or percentage of something, such as the amount of a discount or the amount of interest earned.  

To find the percentage of students who did not receive enough to cover full tuition, we need to divide the difference ($744) by the amount of the scholarship ($3,456). This gives us a percentage of 21.5%.

Rounded to the nearest whole percent, the answer is 22%. This means that 22% of students who received a merit scholarship did not receive enough to cover full tuition.

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Answer:

  6 cm

Step-by-step explanation:

You want the height of a trapezium with bases 17 cm, 35 cm and area 156 cm².

The formula for the are of a trapezium is ...

  A = 1/2(b1 +b2)h

Filling in the given values, we have ...

  156 = 1/2(17 +35)h = 26h

  6 = h . . . . . . . . . . divide by 26

The height of the trapezium is 6 cm.

Answer:

6cm

Step-by-step explanation:

To find:-

Answer:-

We are here given that the area of the trapezium is 156cm² and two of the parallel sides are 17cm and 35cm .We are interested in finding out the height of the trapezium.

The area of the trapezium is given by the formula,

[tex]:\implies \sf Area =\dfrac{1}{2}\times (s_1+s_1)\times h \\[/tex]

where s1 and s2 are the || sides of the trapezium and h is the height of the trapezium.

Now on substituting the respective values in the given formula, we have;

[tex]:\implies \sf 156cm^2 =\dfrac{1}{2} (17cm+35cm)\times h \\[/tex]

[tex]:\implies \sf 156cm^2(2) = 52cm (h) \\[/tex]

[tex]:\implies \sf h =\dfrac{156(2)}{52} cm\\[/tex]

[tex]:\implies \sf \pink{ height = 6 cm }\\[/tex]

Hence the height of the trapezium is 6cm .

Step-by-step explanation:

it was not clear if an average change rate would be sufficient, or if you needed an immediate change rate (as I also don't know if you covered derivatives already or not).

so, it would be helpful, if you could put a message to an answer that was not giving you what you need.

so, here now an answer for an immediate change rate (hopefully that is what you need) :

we have a right-angled triangle.

the direct line of sight (the direct distance between police and red car) is the Hypotenuse (the baseline opposite of the 90° angle).

the 50 ft and 180 ft are the legs.

Pythagoras gives us the length of the Hypotenuse :

Hypotenuse² = 50² + 180² = 2500 + 32400 = 34,900

Hypotenuse = sqrt(34900) = 186.8154169... ft

in general terms let's say x is the distance of the cop to the road, y is the distance on the road to the crossing point with the distance cop to road, and z is the line of sight distance between the red car and the cop (the Hypotenuse).

x² + y² = z²

now, the first derivative of distance is the change of distance = speed.

then dy/dt (= y') is how fast the car is traveling down the road. dx/dt (= x') is how fast the cop is traveling toward the road. and dz/dt (= z') is how fast the distance between the cop and the car is changing.

now, we take the derivative of our equation

x² + y² = z² with respect to time, variable by variable :

d(x² + y² = z²)/dt =

dx²/dx × dx/dt + dy²/dy × dy/dt = dz²/dz × dz/dt

that gives us the equation

2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

x(dx/dt) + y(dy/dt) = z(dz/dt)

from the problem we know x (50 ft), y (180 ft), dz/dt (85 ft/s). we calculated z (the Hypotenuse = sqrt(34900), and since the cop is not moving, we know dx/dt = 0.

and we get

50ft×0ft/s + 180ft×(y') = sqrt(34900)ft×(85)ft/s

we solve for y' (the speed of the car on the road)

y' = sqrt(34900)×85/180 = 88.21839132... ft/s

≈ 88.22 ft/s

and now here the difference for an average change rate over the unrevealed of 1 second :

the radar measured the change of the distance (Hypotenuse) from 1 second ago to now.

so, 1 second ago, the distance was

186.8154169... + 85 = 271.8154169... ft

the 50 ft leg stays the same, but the 180 ft leg was (again via Pythagoras)

271.8154169...² = 50² + leg²

leg² = 271.8154169...² - 50² = 71,383.62088...

leg = 267.1771339... ft

so, the red car traveled

267.1771339... - 180 = 87.1771339... ft/s

as you can see, it is close, but there has to be a difference, as the average change rate is only an approximation to the immediate change rate.

Answer:

P(5,6) = 63

Step-by-step explanation:

Test each point to see which ordered pair maximizes the objective function:

(0,0): p = 3(0) + 8(0) = 0

(2,7): p = 3(2) + 8(7) = 6 + 56 = 62

(5,6): p = 3(5) + 8(6) = 15 + 48 = 63

(8,1): p = 3(8) + 8(1) = 24 + 8 = 32

Hence, (5,6) is the ordered pair that maximizes the objective function.

Answer:

y = 0.8x

Step-by-step explanation:

machine fills 24 jars in 30 seconds , then

fills 1 jar in [tex]\frac{24}{30}[/tex] = 0.8 seconds

thus number of jars filled in x seconds is

y = 0.8x

Answer: 6,160

Step-by-step explanation:

Immediately, without even doing any math, the only logical answer would be 6160. This is because the current tuition is 5,500 and it is increasing so the answer cannot be lower.

However, mathematically you can prove this by turning 12% into a decimal and multiplying it by 5,500. 12% could be converted to .12 and because it is increasing you must add 1, or 100%, since that is what it started with. 5,500 x 1.12 = 6,160.

Answer:

m = 1

Step-by-step explanation:

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points (-1,0) (0,1)

We see the y increase by 1 and the x increase by 1, so the slope is

m = 1

The best interpretation of the confidence interval is We are 95% confident that the difference in proportions for smokers and nonsmokers with CVD in the population is between −0.01 and 0.04 that is option B.

Confidence interval is estimate of Parameter were parameter is difference in proportion of smokers and non smokers in population.

The percentage (frequency) of acceptable confidence intervals that include the actual value of the unknown parameter is represented by the confidence level. In other words, a limitless number of independent samples are used to calculate the confidence intervals at the specified degree of assurance. in order for the percentage of the range that includes the parameter's real value to be equal to the confidence level.

Most of the time, the confidence level is chosen before looking at the data. 95% confidence level is the standard degree of assurance. Nevertheless, additional confidence levels, such as the 90% and 99% confidence levels, are also applied.

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Answer:

2 hours and 56 minutes.

Step-by-step explanation:

To divide 14 hours and 40 minutes by 5, we need to convert everything to minutes first.

14 hours is equal to 14 x 60 = 840 minutes.

So, 14 hours and 40 minutes are equal to 840 + 40 = 880 minutes.

Dividing 880 minutes by 5 gives us:

880 ÷ 5 = 176 minutes

Now, we need to convert the answer back to hours and minutes.

There are 60 minutes in 1 hour, so we can find how many hours are in 176 minutes by dividing by 60:

176 ÷ 60 = 2 with a remainder of 56.

So, the answer is 2 hours and 56 minutes.

Answer: 6y + 6

Step-by-step explanation:

To simplify the expression 8(3/4y -2) + 6(-1/2+4) + 1, we can follow the order of operations (PEMDAS):

First, we simplify the expression within parentheses, working from the inside out:

6(-1/2+4) = 6(7/2) = 21

Next, we distribute the coefficient of 8 to the terms within the first set of parentheses:

8(3/4y -2) = 6y - 16

Finally, we combine the simplified terms:

8(3/4y -2) + 6(-1/2+4) + 1 = 6y - 16 + 21 + 1 = 6y + 6

Therefore, the expression 8(3/4y -2) + 6(-1/2+4) + 1 is equivalent to 6y + 6.

The probability that a vehicle will weigh between 1,946 and 4,455 pounds is 0.8076.

To solve this problem, we need to find the probability that a randomly chosen car weighs between 1,946 and 4,455 pounds. Since weight is uniformly distributed, we know that the probability density function is constant over the entire range of possible values.

First, we need to find the total range of possible values:

Range = maximum weight - minimum weight

Range = 4,665 - 1,557

Range = 3,108

Next, we need to find the range of values that fall between 1,946 and 4,455:

Target range = 4,455 - 1,946

Target range = 2,509

Finally, we can calculate the probability of a randomly chosen car falling within this target range:

Probability = Target range / Range

Probability = 2,509 / 3,108

Probability = 0.8076 (rounded to 4 decimal places)

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The value of (f • p)(-5) is 1885 when functions are given as f(s) = 3s + 2 and p(s) = s³+ 4s.

In mathematics, a function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. It is often represented by an equation or formula, and can be visualized as a graph. Functions are widely used in various areas of mathematics, science, engineering, and other fields to model real-world phenomena and solve problems.

Here,

f(s) = 3s + 2

p(s) = s³+ 4s

To find (f • p)(-5), we need to first find f(-5) and p(-5), and then multiply them together. To find f(-5), we substitute -5 into the function f(s) and simplify:

f(-5) = 3(-5) + 2

= -13

To find p(-5), we substitute -5 into the function p(s) and simplify:

p(-5) = (-5)³ + 4(-5)

= -125 - 20

= -145

Now we can multiply f(-5) and p(-5) together to find (f • p)(-5):

(f • p)(-5) = f(-5) * p(-5)

= (-13) * (-145)

= 1885

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if we make quarterly deposits and invest them at an nominal annual interest rate of 8% compounded quarterly for 40 years, we will have $143,004.54 at the end of the 40 years.

The first step in solving this problem is to calculate the amount of each quarterly deposit. We know that the first deposit is $400, and each subsequent deposit decreases by 1% from the previous deposit. This means that each deposit is 99% of the previous deposit. To calculate the size of each deposit, we can use the following formula:

deposit_ n = deposit_(n-1) * 0.99

Using this formula, we can calculate the size of each quarterly deposit as follows:

deposit_1 = $400

deposit_2 = deposit_1 * 0.99 = $396.00

deposit_3 = deposit_2 * 0.99 = $392.04

deposit_4 = deposit_3 * 0.99 = $388.12

...

We can continue this pattern for 40 years (160 quarters) to find the size of each quarterly deposit.

Next, we need to calculate the future value of these deposits using an nominal annual interest rate of 8% compounded quarterly. We can use the formula for compound interest to calculate the future value:

[tex]FV = PV * (1 + r/n)^(n*t)[/tex]

where FV is the future value, PV is the present value (which is zero since we are starting with deposits), r is the nominal annual interest rate (8%), n is the number of times the interest is compounded per year (4 since we are compounding quarterly), and t is the number of years (40).

We can substitute the values into the formula and solve for FV:

[tex]FV = $400 * (1 + 0.08/4)^(440) + $396.00 * (1 + 0.08/4)^(439) + $392.04 * (1 + 0.08/4)^(4*38) + ... + $1.64 * (1 + 0.08/4)^4[/tex]

After solving this equation, we get a future value of $143,004.54, rounded to two decimal places. This means that if we make quarterly deposits and invest them at an nominal annual interest rate of 8% compounded quarterly for 40 years, we will have $143,004.54 at the end of the 40 years.

This calculation highlights the power of compound interest over long periods of time. By making regular contributions and earning interest on those contributions, our investment grows exponentially over time. It also shows the importance of starting early and consistently contributing to an investment over time in order to achieve long-term financial goals.

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Step-by-step explanation:

Lisa started with $25 on her prepaid debit card. After her first purchase, she had $22.90 left. Therefore, she spent:

$25 - $22.90 = $2.10

We know that the price of the ribbon was 14 cents per yard. To find out how many yards Lisa bought, we can set up an equation:

$2.10 ÷ $0.14/yd = 15 yards

Therefore, Lisa bought 15 yards of ribbon with her prepaid debit card.

Answer:

H. 7

Step-by-step explanation:

Given x² + rx + 12 is equivalent to (x + 3)(x + s), equate the two expressions and expand the right side of the equation:

[tex]\begin{aligned}x^2+rx+12&=(x + 3)(x + s)\\ x^2+rx+12&=x^2 + sx + 3x + 3s\\x^2+rx+12&=x^2 + (s+3)x + 3s\end{aligned}[/tex]

To find the value of r, first find the value of s.

The constant term of the right-hand side must be equal to the constant term of the left-hand side. Therefore:

[tex]\implies 3s = 12[/tex]

Solve for s by dividing both sides of the equation by 3:

[tex]\implies s = 4[/tex]

Compare the coefficients of the terms in x:

[tex]\implies r = s + 3[/tex]

Substitute the value of s into the equation and solve for r:

[tex]\begin{aligned} \implies r &= s + 3\\&= 4 + 3\\&= 7\end{aligned}[/tex]

Therefore, the value of r is 7.

Answer:

[tex]\large\boxed{\sf r = 7 }[/tex]

Step-by-step explanation:

Correct question:- If r and s are constants and r² +rx + 12 is equivalent to (x+3)(x + s), what is the value of r?

Here we are given that , the expression (x+3)(x+s) is equal to r² + rx + 12 .

Firstly, expand the expression (x+3)(x+s) as ,

[tex]\implies (x+3)(x+s) \\[/tex]

[tex]\implies x(x+s)+3(x+s) \\[/tex]

[tex]\implies x^2 + xs + 3x + 3s \\[/tex]

Take out x as common,

[tex]\implies x^2 + (3+s)x + 3s \\[/tex]

Now according to the question,

[tex]\implies x^2 + (3+s)x + 3s = r^2 + rx + 12\\[/tex]

On comparing the respective terms , we get,

[tex]\implies r = 3 + s \\[/tex]

[tex]\implies 3s = 12 \\[/tex]

Solve the second equation to find out the value of s , so that we can substitute that in equation 1 to find "r" .

[tex]\implies 3s = 12 \\[/tex]

[tex]\implies s =\dfrac{12}{3}=\boxed{4} \\[/tex]

Now substitute this value in equation (1) as ,

[tex]\implies r = 3 + s \\[/tex]

[tex]\implies r = 3 + 4 \\[/tex]

[tex]\implies \underline{\underline{ \red{ r = 7 }}} \\[/tex]

and we are done!

The largest printed area of the poster would be when the width and height of the poster are equal. To find the width of the poster, we need to subtract the total area of the margins from the total area of the poster.

The total area of the margins is 6 cm x 4 cm + 4 cm x 6 cm = 88 cm2.

Therefore, 245 cm2 - 88 cm2 = 157 cm2.

We can then use the formula A = W x H, where A is the area and W and H are the width and height of the poster, respectively.

Therefore, 157 cm2 = W x W.

We can solve for W by taking the square root of both sides.

Therefore, W = √157 = 12.5 cm.

Therefore, the width of the poster should be 12.5 cm in order to have the largest printed area.

The set of ordered pairs does not represent a function is the one in the third option.

{(-1,9), (-8, 5), (-1, 3), (-9, 1)}

A relation maps elements from one set, the domain, into elements of other set, the range.

Such that these mappings are of the form (x,y).

A function is a relation where each input is mapped into a single one output, then if you see a relation that has two points with the same value of x and different values of y, then that relation is not a function.

Particularly, if you look at the third option:

{(-1,9), (-8, 5), (-1, 3), (-9, 1)}

You can see that the first and second points have the same input and different outputs, then this is not a function, and that is the correct answer.

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The median is the middle value of a data set when the values are arranged in order from lowest to highest (or highest to lowest). If there is an even number of values, then the median is the average of the two middle values. The median divides the data set into two halves, with half of the values being below the median and half of the values being above the median.

The first quartile, denoted as Q1, is the value that separates the lowest 25% of the data from the rest of the data. The second quartile, denoted as Q2, is the median of the data set. The third quartile, denoted as Q3, is the value that separates the lowest 75% of the data from the rest of the data.

The data-set has in this problem has two-halves of five elements, divided by the number 21, hence the median and the quartiles are given as follows:

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The value stored in 0x10000004 on a big-endian machine is given by 0x00000011.

The word "endianness" refers to the arrangement of bytes as they are stored in computer memory. Endianness is classified as big or little depending on which value is stored first.

The "big end" (the most important item in the sequence) is put first and at the lowest storage address in a big-endian order. The "small end" (the least important item in the sequence) is put first in a little-endian order.

(1)In Big-endian Machine, first byte of multi-byte data will be stored first(at lower memory address)

Address                 Data

0x10000000            0x11

0x10000001            0x22

0x10000002            0x33

0x10000003            0x44

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

lbu $t0, 0($t1)

Load unsigned byte in Register $t0 at address 0x10000000

Here byte at address 0x10000000 is 0x11

$t0 = 0x00000011

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

sw $t0, 0($t2)

Store a word(4 bytes) from Register $t0 to memory address 0x10000004

value stored in 0x10000004 is 0x00000011

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

(2)In Little-endian Machine, last byte of multi-byte data will be stored first(at lower memory address)

Address                 Data

0x10000000            0x44

0x10000001            0x33

0x10000002            0x22

0x10000003            0x11

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

lbu $t0, 0($t1)

Load unsigned byte in Register $t0 at address 0x10000000

Here byte at address 0x10000000 is 0x44

$t0 = 0x00000044

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

sw $t0, 0($t2)

Store a word(4 bytes) from Register $t0 to memory address 0x10000004

value stored in 0x10000004 is 0x00000044.

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(a) θ is a biased estimator for θ.

(b) (α+1)Y/α is an unbiased estimator of θ.

(c) MSE(θ) = αθ^2/[(α+1)^2(α+2)]

(a) To show that θ is a biased estimator for θ, we need to show that E(θ) ≠ θ.

Using the definition of the maximum function, we have

P(θ ≤ y) = P(Y1 ≤ y, Y2 ≤ y, ..., Yn ≤ y) = (F(y))^n

where F(y) is the cumulative distribution function of Y.

Differentiating both sides with respect to y, we get:

f(θ) = n(F(θ))^(n-1)f(θ)

Simplifying, we get

F(θ) = (1/n)^(1/(n-1))

Using this result, we can find the expected value of θ

E(θ) = ∫₀^∞ θf(θ)dθ = ∫₀^θ θαθ^α-1dθ = αθ/(α+1)

Thus, E(θ) ≠ θ, which means that θ is a biased estimator for θ.

(b) To find a multiple of θ that is an unbiased estimator of θ, we can use the method of moments.

We know that the population mean of Y is

μ = ∫₀^θ yf(y)dy = αθ/(α+1)

The sample mean is

Y = (Y1+Y2+...+Yn)/n

Equating these two expressions and solving for θ, we get

θ = (α+1)Y/α

Thus, (α+1)Y/α is an unbiased estimator of θ.

(c) The mean squared error (MSE) of θ can be written as

MSE(θ) = E[(θ - θ)^2]

Expanding the square and using the linearity of expectation, we have

MSE(θ) = E[θ^2] - 2θE[θ] + E[θ]^2

We already know that E[θ] = αθ/(α+1).

To find E[θ^2], we can use the fact that θ = max(Y1,Y2,...Yn)

P(θ ≤ y) = P(Y1 ≤ y, Y2 ≤ y, ..., Yn ≤ y) = (F(y))^n

Differentiating both sides with respect to y, we get

f(θ) = n(F(θ))^(n-1)f(θ)

Using this result, we can find E[θ^2]

E[θ^2] = ∫₀^∞ θ^2f(θ)dθ = ∫₀^θ θ^2αθ^α-1dθ = αθ^2/(α+2)

Substituting these expressions into the MSE formula, we get

MSE(θ) = αθ^2/(α+2) - 2θ(αθ/(α+1)) + (αθ/(α+1))^2

Simplifying, we get

MSE(θ) = αθ^2/[(α+1)^2(α+2)]

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Answer:

57

Step-by-step explanation:

Okay, So we have this person That has 81% average before the quiz. For a course That score includes everything, but the final, which counts for 25% of the course grade, was the best course grade you your friend can earn. Okay, The best course grade given to me makes me. Mhm. 100. So we have .81 times. Actually we'll keep this as 81, times .75 right Plus 100 times. Excuse ME, Time 0.25. This is equivalent to 81 times 0.75. Mhm. This is equivalent to 81 times .75 plus 100 times 0.25, Which equals 85.75%. Now that's for part one. Report to we have what is the minimum score? Turn to 75%. So we have 75 equal to 81 times 0.75 Plus X. Times zero 25. So 81 times 0.75 equals 60.75 75- is value mhm. Is equal to 14.25. So we have 0.25 x. And if we divide 14.25 over 0.25, we isolate X. So the minimum score to get a 75 is a 57

Answer:

r = -2

Step-by-step explanation:

5r + 20/10 = 3r -6/3

5r + 2 = 3r - 2

2r + 2 = -2

2r = -4

r = -2

Answer: r = -2

Step-by-step explanati

Answer:

x = 4

Step-by-step explanation:

If we use A*B = C*D we get 2*6 = 3*x which is 12 = 3x. Dividing both sides by 3 you get 4 = x

The true statements are:

The graph of the function is a parabola.

The graph contains the point (0, 0).

The graph does not contain the point (20, -8).

The quadratic function is f(x) = x^2 - 5x + 12. Here are the statements that are true:

The value of f(-10) = 82:

To find f(-10), we substitute -10 for x in the function:

[tex]$$f(-10) = (-10)^2 - 5(-10) + 12 = 100 + 50 + 12 = 162$$[/tex]

Therefore, the statement "The value of f(-10) = 82" is false.

The graph of the function is a parabola:

Since the highest power of x in the function is 2, the graph of the function will be a parabola. Therefore, the statement "The graph of the function is a parabola" is true.

The graph of the function opens down:

The coefficient of [tex]$x^2$[/tex] in the function is positive (+1), which means the parabola opens upwards. Therefore, the statement "The graph of the function opens down" is false.

The graph contains the point (20, –8):

To see whether the point (20, -8) is on the graph of the function, we substitute x=20 into the function:

[tex]$$f(20) = (20)^2 - 5(20) + 12 = 400 - 100 + 12 = 312$$[/tex]

Since the y-coordinate of the point (20, -8) is not equal to 312, the statement "The graph contains the point (20, –8)" is false.

The graph contains the point (0, 0):

To see whether the point (0, 0) is on the graph of the function, we substitute x=0 into the function:

[tex]$$f(0) = (0)^2 - 5(0) + 12 = 12$$[/tex]

Since the y-coordinate of the point (0, 0) is equal to 12, the statement "The graph contains the point (0, 0)" is true.

Therefore, the true statements are:

The graph of the function is a parabola.

The graph contains the point (0, 0).

The graph does not contain the point (20, -8).

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Therefore, we can use the normal distribution with mean 27.9 and standard deviation 1.67 to approximate the binomial distribution with n = 31 and p = 0.9.

The binomial distribution is characterized by two parameters: the number of trials, denoted by n, and the probability of success on each trial, denoted by p. The probability of obtaining exactly k successes in n trials is given by the binomial probability mass function:

P(k) = (n choose k) * [tex]p^{k}[/tex] * [tex](1-p)^{(n-k)}[/tex],

where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n distinct items.

Given by the question.

Yes, the binomial distribution with n = 31 and p = 0.9 can be approximated by a normal distribution.

The conditions for a binomial distribution to be approximated by a normal distribution are as follows:

n*p >= 10

n*(1-p) >= 10

In this case, n = 31 and p = 0.9, so:

np = 310.9 = 27.9 >= 10

n*(1-p) = 31*0.1 = 3.1 >= 10

Condition 1 is satisfied, but condition 2 is not. Therefore, it is recommended to use a correction factor to improve the approximation.

The correction factor is given by:

[tex]\sqrt[2]{np(1-p)}[/tex]

Substituting the values, we get:

[tex]\sqrt[2]{310.90.1}[/tex]= 1.67

The corrected values for mean and standard deviation are:

mean = np = 310.9 = 27.9

standard deviation = [tex]\sqrt[2]{np(1-p)}[/tex] = 1.67

To learn more about probability:

https://brainly.com/question/11234923

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