BALL
AVENGER
Know When to Hold 'Em
Jack, Brooke and Caroline are chatting after school about a new special edition comic book
that just exploded on the scene. One of the friends owns a signed comic acquired before
the author and artist became famous, while the other two friends are eager to get their
hands on a signed copy. Read the scenarios below and decide if Jack should sell his comic
book to Brooke, make a trade with Caroline or hold on to his signed comic.
Time Required: 10 minutes
LESSON 21: STUDENT ACTIVITY SHEET 1
JACK (SELLER):
Jack collects comic books and last year he went to a local convention to check out some
new artists. He found one he thought was amazing and decided to buy an autographed
comic by him at the show for $5. Fast forward one year and the author is now hugely
popular, with everyone clamoring to buy his comics. Autographed comic books are flying
off local shelves at $25 each and some are even selling for $50 online. Jack's unsure if he
should keep the comic, make a trade or sell it now that it's worth so much money.
Should Jack sell, hold or trade? Why?
BROOKE (BUYER):
Brooke collects rare and original comic books. She loves the new series and needs an
autographed copy of the original comic book to add to her collection. She is willing to pay
whatever it takes to buy it and offers Jack $100 for it.
CAROLINE (TRADER):
Caroline would really like to have a signed copy of the comic book, but there's no way she
can afford to pay $25 or $50 to buy one. She does have three comic books from another
author that she, Jack and Caroline all like a lot. She offers Jack her three comic books for
his autographed comic.

Reduced the equation to one of the standard forms

(a) z²/36 = 4x²/36 + 9y²/36 + 1

(b) x = 2y² + 3z

(c) (x + y + 2z)²

(d) (x - y + z)²

It is represented by an equal sign (=) and is used to solve for unknown values or to describe relationships between variables. Equations can be simple or complex, involving basic arithmetic operations or advanced mathematical concepts like trigonometry and calculus.

An equation can be used to solve for a single unknown value, for example, the equation 2x + 3 = 7 can be solved for x by subtracting 3 from both sides, giving us 2x = 4 and then dividing both sides by 2, giving us x = 2. In this example, x = 2 is the solution to the equation.

(a) The equation z² = 4x² + 9y² + 36 can be reduced to the standard form of an ellipsoid by dividing both sides by 36:

z²/36 = 4x²/36 + 9y²/36 + 1

This is the equation of an ellipsoid with center at the origin and semi-axes of length 3, 3/2, and 6, respectively. The surface is a solid and is classified as an oblate ellipsoid.

(b) The equation x = 2y² + 3z² is a standard parabolic cylinder equation. The surface is a cylinder and is classified as a parabolic cylinder.

(c) The equation 4x² + y² + 4z² - 4y - 24z + 36 = 0 can be reduced to the standard form of an ellipsoid by dividing both sides by 36:

x² + y² + 4z² - 4y - 24z + 36 = 0

= (x + y + 2z)²

This is the equation of an ellipsoid with center at the origin and semi-axes of length 3, 3, and 6, respectively. The surface is a solid and is classified as an oblate ellipsoid.

(d) The equation x² - y² + z² - 4x - 2y - 2z = 0 can be reduced to the standard form of a hyperboloid of two sheets by subtracting 1 from both sides:

x² - y² + z² - 4x - 2y - 2z + 1 = 1

= (x - y + z)²

This is the equation of a hyperboloid of two sheets with center at the point (-2, -1, -1) and semi-axes of length √2, √2, and √6, respectively. The surface is a solid and is classified as a hyperboloid of two sheets.

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After reducing the equation to one of the standard forms, the equations will be (a) z²/36 = 4x²/36 + 9y²/36 + 1, (b) x = 2y² + 3z, (c) (x + y + 2z)², and (d) (x - y + z)²

It is represented by an equal sign (=) and is used to solve for unknown values or to describe relationships between variables. Equations can be simple or complex, involving basic arithmetic operations or advanced mathematical concepts like trigonometry and calculus.

An equation can be used to solve for a single unknown value, for example, the equation 2x + 3 = 7 can be solved for x by subtracting 3 from both sides, giving us 2x = 4 and then dividing both sides by 2, giving us x = 2. In this example, x = 2 is the solution to the equation.

(a) The equation z² = 4x² + 9y² + 36 can be reduced to the standard form of an ellipsoid by dividing both sides by 36:

z²/36 = 4x²/36 + 9y²/36 + 1

This is the equation of an ellipsoid with center at the origin and semi-axes of length 3, 3/2, and 6, respectively. The surface is a solid and is classified as an oblate ellipsoid.

(b) The equation x = 2y² + 3z² is a standard parabolic cylinder equation. The surface is a cylinder and is classified as a parabolic cylinder.

(c) The equation 4x² + y² + 4z² - 4y - 24z + 36 = 0 can be reduced to the standard form of an ellipsoid by dividing both sides by 36:

x² + y² + 4z² - 4y - 24z + 36 = 0

= (x + y + 2z)²

This is the equation of an ellipsoid with center at the origin and semi-axes of length 3, 3, and 6, respectively. The surface is a solid and is classified as an oblate ellipsoid.

(d) The equation x² - y² + z² - 4x - 2y - 2z = 0 can be reduced to the standard form of a hyperboloid of two sheets by subtracting 1 from both sides:

x² - y² + z² - 4x - 2y - 2z + 1 = 1

= (x - y + z)²

This is the equation of a hyperboloid of two sheets with center at the point (-2, -1, -1) and semi-axes of length √2, √2, and √6, respectively. The surface is a solid and is classified as a hyperboloid of two sheets.

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Answer:

B. 60

Step-by-step explanation:

Since the digits 1, 1, 1, 2, 2, 3 are repeated, we use the concept of permutations with repetition. In this case, we can choose the first digit to be any one of the three, the second digit to be any one of the three, and so on. Therefore, the total number of distinct 6-digit numbers that can be made is 3 × 3 × 3 × 3 × 3 × 3 = 729.

However, we need to exclude the numbers that start with a 0 (i.e., numbers like 011112 or 001122). Therefore, the final answer is 729 - 3 = 726, which is the number of 6-digit numbers that do not start with a 0. The number of 6-digit numbers that start with a 0 is 3 (111120, 111202, 112102), so the total number of distinct 6-digit numbers is 726 - 3 = 60.

Answer:

Step-by-step explanation:

angle DGF

Answer: $14.22

Step-by-step explanation: 237*0.06

Answer:

1/36

Step-by-step explanation:

above is the answer

The answers for each polynomial are:

a) 4x³ - 8: degree is 3, and the number of terms is 2.

b) 2x⁴ - 7x² - 5x + 1: degree is 4, and there are 4 terms.

c) x³ +5x² + 28 - x:   standard form is: x³ +5x²  - x + 28 And the leading coefficient is 1.

d) 24 - x³ + x:   standard form is:  -x³ + x + 24 and the leading coefficient is -1.

The degree of a polynomial is equal to the maximum exponent in the variable, and the number of terms is just the number of "elements" separated by the symbols + or -.

For the first one:

4x³ - 8

The degree is 3, and the number of terms is 2.

For the second one:

2x⁴ - 7x² - 5x + 1

The degree is 4, and there are 4 terms.

In the next part you need to write them in standard form and write the leading coefficient. The leading coefficient is the coefficient on the term with the largest exponent.

For the first one:

x³ +5x² + 28 - x

The standard form is:

x³ +5x²  - x + 28

And the leading coefficient is 1.

And the last one is:

24 - x³ + x

The standard form is:

-x³ + x + 24

The leading coefficient is -1.

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9¹⁰ can be expressed as base 4 with 22 and 15¹⁰ can be expressed as 33.

An expression is combination of variables, numbers and operators.

To change from base 10 to base 4, we need to express each number in terms of powers of 4. Here's how you can do this for each number:

9¹⁰ in base 4:

9 in base 10 can be expressed as 22

Base 4

So, (22)¹⁰ = 4¹⁰ x 2¹⁰ = 1048576 x 1024

= 1073741824 in base 10.

15¹⁰ in base 4

15 in base 10 can be expressed as 33 in base 4

Hence, 9¹⁰ can be expressed as base 4 with 22 and 15¹⁰ can be expressed as 33.

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Answer:

300 cm^2

Step-by-step explanation:

An area of a triangle is b*h/2.

Therefore, plug in the base and height into the equation.

24cm * 25cm / 2 = area

600cm^2 / 2 = area

300cm^2 = area

After resolving [tex](c+3)^{2}[/tex] , c = -3, and the approach is an example of the distributive property of equations.

Its simplest form has been thought to be the figure is probably smallest equivalent fraction. How or when to determine the basic form. Look for shared factors throughout the denominator and numerator. A fractional number can sometimes be checked to discover if it is a power of 2. A statement having two equal sides and an equal sign in the middle is referred to as a mathematical equation. Every variable's degree courses are listed in descending order as in general form of any equation. The formula for a linear equation is x + b = 0. The general form of a three separate linear equation is an x + b y + c = 0. (or something similar). Either conditioned equations or identities are categories for mathematics. An identity holds true for whatever value of the variables.

Given,

[tex](c+3)^{2}[/tex]

= by using formulas of [tex](a+b)^{2}[/tex]

by distributive property

[tex]c^{2}[/tex] + [tex]3^{2}[/tex] + 6c

= [tex]c^{2}[/tex] + 9 + 6c

= [tex]c^{2}[/tex] + 6c + 9 = 0

[tex]c^{2}[/tex] + 3c + 3c + 9 = 0

= c ( c + 3 ) + 3 ( c + 3)

= (c + 3 )(c+ 3)

c = - 3

After resolving [tex](c+3)^{2}[/tex], c = -3, and the approach is an example of the distributive property of equations.

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The value of x in the given paralellogram is 12.

What is parallelogram?

"It is a quadrilateral with parallel opposite sides."

"The opposite angles of parallelogram are equal. "

For given question,

Two of the angles in a parallelogram measure 8x° and 7x°

The measure of an angle opposite to 8x° is also 8x°

In a similar manner, an angle opposite to 7x° is measured as 7x°.

So, we have the four angles of parallelogram as 8x°, 7x°, 8x°, 7x°

We are know that the total angle of a quadrilateral is 360°.

⇒ 8x° + 7x° + 8x° + 7x° = 360°

⇒ (5 + 7 + 5 + 7) x = 360

⇒ 30x = 360

⇒ x = 12

Therefore, the value of x is 15.

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Making A the subject of the equation V = APT gives A = V/(PT)

An equation is an expression that contains numbers and variables linked together by mathematical operations of addition, subtraction, multiplication, division and exponents. An equation can either  be linear, quadratic, cubic, depending of the degree of the variable.

Given the equation:

V = APT

We need to make A the subject of the formula.

Firstly, divide both sides of the equation by P:

V/P = APT/P

AT = V/P

Secondly divide both sides of the equation by T:

(AT)/T = (V/P)/T

A = V/(PT)

The value is A = V/(PT)

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Maryann needs to run 2 more laps to run on Sunday .

Lap can be referred as One complete trip around a race track οr from one end of a swimming pool to the other: Fοr example, Each lap of the track is 400 meters.

Given:

Maryann runs : 3/4-mile track

She runs  on Saturday: 8 1/2 laps = 17/2

Total miles to cover = 10 1/2 = 21/2

more laps does Maryann need to run on Sunday

= Total miles to cover - She runs  on Saturday

= 21/2 - 17/2

= 4/2

= 2

Hence , Maryann needs to run 2 more laps to run on Sunday .

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Answer:

C (51)

Explanation

Number of acres more there at Navajo Monument than at George Washington Carver Monument is 120.

Find the area that its outer surfaces possess. Sum of all those surfaces' area is the surface area of the considered object.

Given;

The area of navajo monument= 360acres

The area of George Washington Carver Monument = 240 acres

Now

More area navajo monument has =area of navajo monument -  area of George Washington Carver Monument

=360-240

=120

Therefore, the extra area navajo monument has is 120 acres.

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The value of 2:3 can be written as a fraction: 2/3.

What is the ratio?

In mathematics, a ratio is a comparison of two quantities, often expressed as a fraction or using the colon (:) symbol. Ratios are used to indicate the relative size or amount of two or more things.

For example, the ratio of the length of a rectangle to its width can be expressed as a fraction, where the numerator represents the length and the denominator represents the width.

The ratio of hydrogen atoms to nitrogen atoms is 2:3.

This means that for every 2 hydrogen atoms, there are 3 nitrogen atoms in the container. The value of the ratio 2:3 is a way of expressing the relative quantity of hydrogen atoms to nitrogen atoms in a simplified form. It can also be written as a fraction: 2/3.

Hence, the value of 2:3 can be written as a fraction: 2/3.

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Answer: (x, y) = (5/4, 2).

The given system of equations is:

4x - y = 3

3x + y = 4

We can solve for x and y using either substitution or elimination method.

By substitution, we can solve for y in one of the equations and substitute that value into the other equation to solve for x.

Starting with 4x - y = 3, we can add y to both sides to isolate x:

4x = 3 + y

Then, we can substitute this expression for y into the second equation:

3x + (3 + y) = 4

Expanding the right side, we get:

3x + 3 + y = 4

3x + 3 = 4 - y

3x = 1 - y

x = (1 - y)/3

Now, we can substitute the expression for x into the first equation:

4((1 - y)/3) - y = 3

4(1 - y)/3 = 3 + y

4 - 4y/3 = 3 + y

1 - 4y/3 = y + 3

-3y/3 = -2

y = 2

Finally, we can substitute the value of y into one of the original equations to find x:

4x - 2 = 3

4x = 5

x = 5/4

So, the solution to the system of equations is (x, y) = (5/4, 2).

The factor of 1 - 2.25x⁸ is (1 + 1.5x⁴) (1 - 1.5x⁴).

The correct option is C.

Factorization in mathematics is the process of dividing a large number into smaller ones that, when multiplied together, give you the original number. Factorization is the process of dividing a number into its components or divisors.

Given:

An expression,

1 - 2.25x⁸.

Simplifying to the factored form.

1² - (1.5x⁴)²

= (1 + 1.5x⁴) (1 - 1.5x⁴)

Therefore, (1 + 1.5x⁴) (1 - 1.5x⁴) is the factored form.

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Answer: The annual percentage yield (APY) is the total amount of interest earned on an investment over a year, expressed as a percentage of the initial investment. The annual interest rate is simply the interest rate per year, expressed as a percentage.

To convert the APY to the annual interest rate, we can divide the APY by the number of times the interest is compounded in a year. If the interest is compounded monthly, then the number of times the interest is compounded in a year is 12.

So, the annual interest rate for the account is given by:

Annual interest rate = APY / number of times compounded in a year

= 5.9% / 12

= 0.49%

Therefore, the annual interest rate for the account is 0.49%.

Step-by-step explanation:

The largest N such that the probability that there are at least N survivors at age 95 is at least 90% is approximately 136.43.

The normal approximation can be used to estimate the number of survivors at a given age when the mortality follows the Standard Ultimate Life Table. The normal approximation assumes that the number of survivors follows a normal distribution.

Let X be the number of survivors at age 95. Then, the mean (μ) and standard deviation (σ) of X can be calculated using the following formula:

μ = 4,000 * e⁽⁻⁰.⁰⁷³⁶⁽⁹⁵⁻²⁵⁾⁾

σ = √(4,000 * (1 - e⁽⁻⁰.⁰⁷³⁶⁽⁹⁵⁻²⁵⁾⁾) * e⁽⁻⁰.⁰⁷³⁶⁽⁹⁵⁻²⁵⁾⁾)

Next, we can find the value of N such that P(X >= N) >= 0.9 using the standard normal distribution.

z = (N - μ) / σ

P(X >= N) = 1 - P(X < N) = 1 - Φ(z)

where Φ(z) is the cumulative distribution function of the standard normal distribution.

We want to find the largest N such that P(X >= N) >= 0.9, so we need to find the smallest z such that Φ(z) <= 0.1. The value of z can be found using a standard normal table or a calculator with normal distribution functions.

With a calculator, we get that z = 1.28. So, N = μ + z * σ.

Plugging in the values we calculated earlier, we get:

N = 4,000 * e⁽⁻⁰.⁰⁷³⁶⁽⁹⁵⁻²⁵⁾⁾ + 1.28 * √(4,000 * (1 - e⁽⁻⁰.⁰⁷³⁶⁽⁹⁵⁻²⁵⁾⁾) * e⁽⁻⁰.⁰⁷³⁶⁽⁹⁵⁻²⁵⁾⁾)

N = 4,000 * e⁽⁻⁰.⁰⁷³⁶⁽⁷⁰⁾⁾ + 1.28 * √(4,000 * (1 - e^⁽⁻⁰.⁰⁷³⁶⁽⁷⁰⁾) * e^⁽⁻⁰.⁰⁷³⁶⁽⁷⁰⁾⁾

N = 4,000 * 0.0337 + 1.28 * √(4,000 * (1 - 0.0337) * 0.0337)

N = 132.8 + 1.28 * √(4,000 * 0.966 * 0.0337)

N = 132.8 + 1.28 * √(13.97376)

N = 132.8 + 3.63323

N ≈ 136.43

So, the largest N such that the probability that there are at least N survivors at age 95 is at least 90% is approximately 136.43.

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The simplified expression for this problem is given as follows:

[tex]43\sqrt{5} - 6[/tex]

The expression for this problem is defined as follows:

[tex]30\sqrt{5} + 13\sqrt{5} - 6[/tex]

To simplify the expression, we combine the like terms, which are the terms that have the same rational, thus:

[tex]30\sqrt{5} + 13\sqrt{5} = 43\sqrt{5}[/tex]

Combining along with the subtraction by 6, the complete simplified expression is given as follows:

[tex]43\sqrt{5} - 6[/tex]

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The inequality describing the given problem is x/5 ≥ 2.8 and Troy should at least jog for 14 miles in total.

A relationship between two expressions or values that are not equal to each other is called inequality.

Given that, Troy set a goal for the total distance he wants to jog this week. He divided his goal evenly among the 5 jogs he planned for the week. Troy determined that each jog should be at least 2.8 miles to meet his goal.

x represent how many miles Troy wants to jog in all. We need to find an inequality describes the problem.

Since, he has divided his goal into 5 jogs and in total he did x jogs, therefore, jog for 1 day = x/5

Now, he set that each jog should be at least 2.8 miles to meet his goal.

That mean, he can jog 2.8 miles or more than it but not less than it,

Therefore, inequality will be,

x/5 ≥ 2.8

Now, finding x,

Multiply both sides by 5,

x ≥ 14

Hence, the inequality describing the given problem is x/5 ≥ 2.8 and Troy should at least jog for 14 miles in total.

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The system of equations is solved by graphing as shown in the image, where the solution is (1, 2).

A system of equations can be solved by graphing each line that represents each equation, then find the point where they both intercept.

Thus, the coordinates of the point where the two linear equations intercept will be the solution of the system of equations.

Given the system, 9x - 7y = -5 is the red line while -3x + 7y = 11 represents the blue line. Both intercept at (1, 2), which is the solution of the equations.

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The theoretical probability of rolling a 4 on the die is 0.0647

The binomial distribution is a type of probability distribution that predicts the likelihood of obtaining one of two outcomes given a set of inputs. It summarizes the number of tries where each trial has the equal chance of producing the same result.

The formula for Binomial Distribution is given by

P ( x ) = [ n! / ( n - x )! x! ] pˣqⁿ⁻ˣ  

where

n = number of trials

x = number of successes

p = probability of getting a success in one trial

q = probability of getting a failure in one trial

q = 1 - p

Given data ,

Let the number of trials n = 20

Let the probability of getting a 4 on the die be = 1/6

So , the value of p = 1/6

And , the probability of getting a failure q = 5/6

So , theoretical probability of rolling a 4 on this die = P ( x = 4 )

And , the Binomial Distribution is given by

P ( x = 6 ) = [ 20! / ( 20 - 6 )! 6! ] ( 1/6 )⁶ ( 5/6 )¹⁴

On simplifying the equation , we get

P ( x = 6 ) = [ 20 x 19 x 18 x 17 x 16 x 15 / 6 x 5 x 4 x 3 x 2 x 1 ) ( 1/6 )⁶ ( 5/6 )¹⁴

On further simplification , we get

P ( x = 6 ) = 0.0647

P ( x = 6 ) =  6.047 %

Hence , the probability of rolling a 4 is 0.0647

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The density function of Y = | X-1 | is given by:

f_Y(y) = {f_X(y + 1), if 0 < y < 1;

        {f_X(1 - y), if 1 < y < 2;

        {0, otherwise.

Continuous distribution is a type of probability distribution that describes the likelihood of a variable taking on any value within a given range, rather than just specific values

If X ~Unif (-1,0), then the density function of Y is given by:

f_Y(y) = {1/2, if 0 < y < 1;

        {1/2, if 1 < y < 2;

        {0, otherwise.

Therefore, Y ~Unif (0,2).

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The difference in height of the two checkpoints is 3487 feet.

The term above mean sea level (AMSL) refers to the elevation (on the ground) or altitude (in the air) of any object, relative to the average sea level datum.

Given here: The table containing the altitude of different checkpoints in a race.

Height above mean sea level is a measure of the vertical distance (height, elevation or altitude) of a location in reference to a historic mean sea level taken as a vertical datum.

The altitude of checkpoint 1 is -164 and Checkpoint 2 as 3487

Thus Difference in the height of the two checkpoints is

3487-(-164)

3651 feet lower.

If the top of a hill rises 291 feet above checkpoint 1 we get

-164+291=127 feet

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The required events E and H are not mutually exclusive, because they can both occur at the same time.So  D - No, because P(E∩H) = 0.20 is correct.

The events E and H are not mutually exclusive, because they can both occur at the same time. In other words, a student can have both brown eyes and brown hair.

The probability that a student has both brown eyes and brown hair, P(E∩H), can be found using the formula:

P(E∩H) = P(E) + P(H) - P(E or H)

where P(E) is the probability of having brown eyes, P(H) is the probability of having brown hair, and P(E or H) is the probability of having either brown eyes or brown hair.

Given that:

P(E) = 0.35

P(H) = 0.45

P(E or H) = 0.60

So,

P(E∩H) = 0.35 + 0.45 - 0.60 = 0.20

So, the answer is D - No, because P(E∩H) = 0.20.

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Answer:

the inequality y ≥ 0.8x + 1.5 accurately describes the relation between the duration of the hike (x) and the amount of water needed (y). This inequality means that for any duration of the hike (x), the amount of water needed (y) must be greater than or equal to 0.8x + 1.5.

Answer:

Step-by-step explanation:

3 liters of water for 4 hours of hiking.

The likelihood that a stranger on the street has an IQ below 98 is 0.4470, or 44.70%.

The probability that an event will happen or a claim will be correct is measured by statistical mechanics, a branch of mathematics. The possibility of an occurrence is a positive integer and 1, where about 0 represents how likely the event is to occur and value of 1 indicates certainty. A probability is a numerical illustration of the possibility that a specific activity will take place. Probabilities can also be expressed using percentages extending from 0% to 100% or from 0 to 1. the ratio of the number of outcomes to the proportion of occurrences in a comprehensive set of equally likely choices that lead to a specific occurrence.

given

This is Z's pvalue for X = 98. So

Z = 98 -100 / 15

Z = -0.133

Z = -0.133 pvalue of 0.4470

The likelihood that a stranger on the street has an IQ below 98 is 0.4470, or 44.70%.

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The number of farms whose land and building values are between $1,200 and $1,400 is given as follows:

50 farms.

The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:

Considering the mean of $1,300 and the standard deviation of $100, measures between $1,200 and $1,400 are within one standard deviation of the mean, hence the percentage is of 68%.

The total amount is of 74 farms, hence the amount relative to 68% of 74 is of:

0.68 x 74 = 50 farms. (rounded).

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The combination of the difference of the functions (g - f)(x), where; f(x) = 8·x² + 16·x + 6 and g(x) = x³ - 3·x² - 9 is; x³ - 11·x² - 16·x - 15

The correct option is option C

C) x³ - 11·x² - 16·x - 15

The combinations of functions can be found as follows;

Addition; (f + g)(x) = f(x) + g(x)

Subtraction; (f - g) = f(x) - g(x)

Multiplication; (f·g) = f(x) × g(x)

Division; (f/g)(x) = f(x)/g(x)

What is a function?

A function is a rule or law that maps an input variable into an output variable such that each element in the set of input variable is mapped to exactly one element in the set of the output variables.

The f(x) = 8·x² + 16·x + 6

g(x) = x³ - 3·x² - 9

Therefore;

(g - f)(x) = g(x) - f(x)

(g - f)(x) = x³ - 3·x² - 9 - (8·x² + 16·x + 6) = x³ - 11·x² - 16·x - 15

The correct option is therefore option C

C) x³ - 11·x² - 16·x - 15

Learn more on combining functions here: https://brainly.com/question/12000899

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