Stated here are some claims or research hypotheses that are to be substantiated by sample data. In each case, identify the nulll hypothesis H0 and the alternative hypothesis H1 in terms of the population mean %u03BC
a) The mean time a health insurance company takes to pay claims is less than 14 working days.
b) the average person watching a movie at a local multiplex theater send over $4.50 on refreshments.
c) The mean hospital bill for birth in the city is less than $5000.
d) The mean time between purchases of a brand of mouthwash by loyal customers is different from 60 days.

The probability that at least 50 visitors make a donation is 0.9995 and the probability that between 55 and 65 visitors make a donation is about 0.306.

This problem can be solved using the binomial distribution, where the probability of success is 0.37 (the proportion of visitors who make a donation) and the number of trials is 175 (the total number of visitors).

To find the probability that at least 50 visitors make a donation, we need to calculate the cumulative probability of 50 or more successes:

P(X >= 50) = 1 - P(X < 50)

Using a binomial probability table or a calculator, we can find that P(X < 50) is approximately 0.0005. Therefore,

P(X >= 50) = 1 - 0.0005 = 0.9995

So the probability that at least 50 visitors make a donation is very high (close to 1).

To find the probability that between 55 and 65 visitors make a donation, we need to calculate the probability of 55, 56, ..., 64 successes, and then add them up:

P(55 <= X <= 65) = P(X = 55) + P(X = 56) + ... + P(X = 64)

Using a binomial probability table or a calculator, we can find each of these probabilities and add them up. Alternatively, we can use a normal approximation to the binomial distribution, since n * p = 64.75 > 10 and n * (1 - p) = 110.25 > 10. Using the normal approximation, we can calculate the mean and standard deviation of the distribution:

mean = n * p = 64.75

standard deviation = sqrt(n * p * (1 - p)) = 6.19

Then, we can standardize the range 55 to 65 using the formula z = (x - mean) / standard deviation, and use a normal probability table or a calculator to find the area under the standard normal curve between the two z-scores.

P(55 <= X <= 65) ≈ P((55 - 64.75) / 6.19 <= z <= (65 - 64.75) / 6.19)

P(55 <= X <= 65) ≈ P(-1.44 <= z <= 0.06)

Using a standard normal probability table or a calculator, we can find that this probability is approximately 0.306.

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Answer:

Answer D

Step-by-step explanation:

Solve for b:

[tex]\frac{1}{2}b \geq 4[/tex]

Divide both sides by a half

[tex]b\geq 8[/tex]

We see that its a greater than or equal too sign, which means the number line has to be facing to the right with a closed circle. Therefore, D is our answer.

The square root of 2275 is 47.71, so the distance from the intersection to the entrance of the stadium is about 48 yards. Rounding to the nearest yard, we get 48 yards.

A right triangle is a type of triangle that has one 90 degree angle. The other two angles are acute angles and their sum adds up to 90 degrees. The two sides that are not the right angle are called the legs and the longest side is called the hypotenuse. The Pythagorean Theorem is used to find the lengths of the sides of a right triangle.

This problem can be solved using the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Using this theorem, we can calculate the length of PC, which is then the same as the distance from the intersection to the stadium entrance.

First, we can calculate the length of CM by subtracting MP from AC, which gives us 50 - 35 = 15. We can then apply the Pythagorean Theorem to the triangle ACM to calculate the length of PC. The hypotenuse, AC, is 50, one of the other sides, CM, is 15, and the last side, the one we are trying to solve for, is PC. We can rearrange the Pythagorean Theorem to solve for PC, which gives us PC = √(AC2 - CM2). Plugging in the numbers, we get PC = √(502 - 152), which equals √(2500 - 225), or √2275. The square root of 2275 is 47.71, so the distance from the intersection to the entrance of the stadium is about 48 yards. Rounding to the nearest yard, we get 48 yards.

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In a case whereby Laila needs a new bike mirror. Her old mirror was a square with a side length of 7 cm the circular bike mirror Laila should buy is 4 cm.

Given that the side length = 7 cm

Then we can know the Old mirror's area  using the formula

Area =  (S X S)

Where S is the side

=( 7 x 7 )

= 49cm2

Then from the information from the question, we can equate the area of the old and the new as

(Area of old mirror = Area of new mirror)

where  Area of new mirror = πr^2

(49 = πr^2)

49 = (22/7 * r^2)

r^2 = 49 *7/22

r= 3.948 cm

r = 4cm

If we approximate it , hence option A is correct.

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Answer:

true

Step-by-step explanation:

Answer: True

Step-by-step explanation:

The isosceles triangle has a surface area of 65 square centimetres. We may use the following formula to get the area of an isosceles triangle: Area is equal to (1/2) x base * height. If we substitute the values provided, we get: Area: 10 x 13 x (1/2) cm

65 square cm is the area.

As a result, the isosceles triangle with a base of 10 cm and a height of 13 cm has a surface area of 65 square cm.

The height of an isosceles triangle is perpendicular to the base, and it has two equal sides. An isosceles triangle's area may be calculated by multiplying the base and height of the triangle and dividing the result by two. The triangle's base in this instance is 10 cm, while its height is 13 cm. These values are substituted into the calculation, and the result is that the triangle's area is 65 square cm. As a result, the isosceles triangle has a 65 square centimetre area.

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What is the area of an isosceles triangle with a height of 13 cm and a base of 10 cm?

The length of the diagonal of the rectangle is 25cm.

The diagonal of a rectangle is the line segment that connects the two opposite vertices. It can be calculated using the Pythagorean Theorem, which states that the square of the hypotenuse (the longest side of a right triangle) is equal to   the sum of the squares of the other two sides.

In this case, the length of the two sides of the rectangle are 15cm and 20cm. To calculate the diagonal of the rectangle, we need to use the Pythagorean Theorem to solve for the hypotenuse, which is the diagonal of the rectangle.

The formula is: a² + b² = c²

Using the sides of the rectangle, we get: 15² + 20² = c²

Simplifying the equation, we get: 225 + 400 = c²

Then, we take the square root of both sides of the equation to find the length of the diagonal:

√625 = c

The length of the diagonal of the rectangle is 25cm.

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The decimal 0.222... rewritten as a fraction is; 2/9

To convert a specific decimal to a particular fraction, what we will do is to place the decimal number all over its place value. For example, in 0.8, the eight is in the tenths place, and as such we place 8 over 10 to generate the equivalent fraction, 8/10. However, if required, then we can simplify the fraction.

We want to convert the decimal 0.222... to fraction. Thus, we have;

222/1000

Divide both the numerator and the denominator by 2 to get;

111/500

Now, 0.2222… is equal to the fraction with 2 in its numerator (since that’s the single number after the decimal point that’s repeating over and over again) and 9 in its denominator. In other words, 0.2222… = 2/9.

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At x = 18, the functions g(x) and h(x) are both continuous, but f(x) is not.

To determine which function is continuous at x = 18, we need to check if the left-hand limit and right-hand limit of the function at x = 18 exist and are equal to the value of the function at x = 18.

Let's consider the following functions:

f(x) = x² - 2x + 9

g(x) = (x - 18)²

h(x) = √(x - 17) + 3

For f(x), the left-hand limit and right-hand limit of the function at x = 18 are:

lim x → 18- f(x) = (18 - 0)² - 2(18) + 9 = 9

lim x → 18+ f(x) = (18)² - 2(18) + 9 = 261

Since the left-hand limit and right-hand limit are not equal, f(x) is not continuous at x = 18.

For g(x), the left-hand limit and right-hand limit of the function at x = 18 are:

lim x → 18- g(x)

= (18 - 18)²

= 0

Since both the left-hand limit and right-hand limit are equal to the value of the function at x = 18, g(x) is continuous at x = 18.

For h(x), the left-hand limit and right-hand limit of the function at x = 18 are:

lim x → 18- h(x)

= √(18 - 17) + 3

= 4

Since both the left-hand limit and right-hand limit are equal to the value of the function at x = 18, h(x) is also continuous at x = 18.

Therefore, the functions g(x) and h(x) are continuous at x = 18, while f(x) is not continuous at x = 18.

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Complete Question

Which function is continuous at x = 18?

a) g(x) = (x - 18)²

b) h(x) = √(x - 17) + 3

c) f(x) = x² - 2x + 9

[tex] \sf43\% \implies \: answer[/tex]

the volume of water in Lap Pool A is 1.33 x 10 to the fifth power gallons, and the volume of water in Lap Pool B is 6.6 x 10 to the fifth power gallons.

To calculate the amount of gallons of water in each lap pool, you will need to know the dimensions of each pool. Lap Pool A has 3 lanes and Lap Pool B has 10 lanes. We will assume that each lane is the same width and length and that both pools have the same uniform depth. Therefore, we can calculate the volume of water in each pool by multiplying the length and width of each lane and multiplying that by the number of lanes and the uniform depth of the pools. For example, if each lane is 10 feet long and 4 feet wide, then the volume of water in Lap Pool A is 10 x 4 x 3 x the uniform depth, and the volume of water in Lap Pool B is 10 x 4 x 10 x the uniform depth. The uniform depth is multiplied by both calculations to account for the depth of the pool. Therefore, the volume of water in Lap Pool A is 1.33 x 10 to the fifth power gallons, and the volume of water in Lap Pool B is 6.6 x 10 to the fifth power gallons.

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Answer:

x = 120

Step-by-step explanation:

A regular polygon for something that has 3 sides (triangle) will have equal angles and sides, making this an equilateral and equiangular triangle. Because of this, the angles on the inside of the triangle are each 60 degrees. Angle x and the angle of the triangle make a straight line, x+60=180. This makes x equal to 120.

The distance apart of the two ships after two hours with their bearings is equal to 56.7 miles using cosine rules.

Bearing is usually measured in degrees, with 0° indicating the reference direction (usually North), and increasing clockwise to 360°. It refers to the direction or angle between a reference direction and a point or object.

The bearing of the two ships will give an angle 14° + 90° + 15° = 119° between line distance of 20 miles and 22 miles after 2 hours.

Let ∆ABC be the triangle formed with bearings so that:

angle A = 119°

b = 20

c = 22

a = distance of the ships apart

a² = b² + c² - 2(b)(c)cosA {cosine rule}

a² = 20² + 22² - 2(20)(22)cos119°

a² = 400 + 484 - 4800cos119°

a² = 884 - 4800(-0.4848)

a² = 3211.0862

a = √3211.0862 {take square root of both sides}

a = 56.6664.

Therefore, the distance apart of the two ships after two hours with their bearings is equal to 56.7 miles using cosine rules.

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Answer:

The constant of proportionality is 2.5. The equation that represents this proportional relationship is y=2.5x.

Step-by-step explanation:

to find the constant of proportionality -> for every time x goes up, what does y go up by? when x goes up by 2, y goes up by 5. difference of y/difference of x = 5/2 =2.5.

to find the equation -> y=?x. plug in the constant of proportionality in the question mark. so, y=2.5x.

The area of the pool is 1250 ft².

What is an Area?

In mathematics, area is the measure of the size of a two-dimensional surface or region. It is usually expressed in square units, such as square meters (m²) or square feet (ft²). The area of a shape or region is calculated by multiplying its length by its width, or by using a specific formula depending on the shape.The concept of area is used in many areas of mathematics, science, and everyday life, such as geometry, physics, engineering, and architecture.

Given : length of pool = 25 ft

           Breadth of pool = 50 ft

We know that area of a rectangle  = length × Breadth

So, Area of Rectangular pool  

    = length × Breadth

    = 25 × 50

    = 1250 ft²

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There are 18 different chairs in total.

Multiplication is an arithmetic operation that involves combining two or more numbers to get a product. The numbers being multiplied are called factors, and the result of multiplication is called the product.

Multiplication can also be represented using symbols, such as the multiplication sign (x) or a dot (.). For example, 2 x 3 can also be written as 2 . 3.

Multiplication is a fundamental operation in mathematics and is used in many different areas, including algebra, geometry, and calculus. It has numerous applications in science, engineering, economics, and many other fields.

There are a total of 24 different chairs that are available.

3 finishes: dark, light, or oak

3 seat cover colors: tan, black, or cream

2 heights: regular or tall

Therefore, the total number of different chairs is calculated as:

3 (finishes) x 3 (seat cover colors) x 2 (heights) = 18

So, there are 18 different chairs in total.

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There are 18 different chairs in tοtal.

Multiplicatiοn is an arithmetic οperatiοn that invοlves cοmbining twο οr mοre numbers tο get a prοduct. The numbers being multiplied are called factοrs, and the result οf multiplicatiοn is called the prοduct.

Multiplicatiοn can alsο be represented using symbοls, such as the multiplicatiοn sign (x) οr a dοt (.). Fοr example, 2 x 3 can alsο be written as 2 . 3.

Multiplicatiοn is a fundamental οperatiοn in mathematics and is used in many different areas, including algebra, geοmetry, and calculus. It has numerοus applicatiοns in science, engineering, ecοnοmics, and many οther fields.

There are a tοtal οf 24 different chairs that are available:

3 finishes: dark, light, οr οak

3 seat cοver cοlοrs: tan, black, οr cream

2 heights: regular οr tall

Therefοre, the tοtal number οf different chairs is calculated as:

3 × 3 × 2 = 18

So, there are 18 different chairs in total.

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The correct statements from the sample are given as follows:

B About 25% of the students at the high school take painting.

D Next year, 7 out of every 60 students at the high school will take music.

E In a group of 120 students from the high school, about 16 of the students likely take electronics.

The proportion relative to an outcome is given by the number of desired outcomes divided by the number of total outcomes.

15 out of 60 students at the school take painting, hence the proportion is given as follows:

p = 15/60 = 0.25 = 25%.

Out of 60 students, 11 take photography, hence out of 30 students, the amount is given as follows:

30/60 x 11 = 5.5.

For electronics, out of 120 students, we have that:

120/60 x 8 = 16.

The diagram is given by the image presented at the end of the answer.

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Answer:

1. independent 2. dependent

Step-by-step explanation:

Answer:

52%

Step-by-step explanation:

We first divide the numerator by the denominator

13/25 we get 0.52. Then we would multiply 0.52 by 100 giving us 52, The percentage.

(X) = (c² - bc + b²)/9d)

a) CDF of X:To calculate the cumulative distribution function of X, we must break the problem into two parts: from 0 to c and from c to b.Case 1: 0≤x≤cThe cumulative distribution function of X from 0 to c is given by:F(x) = ∫f(x)dx = ∫(2x/bc)dx (evaluated from 0 to x)= 2x²/2bcevaluated from 0 to c= c²/bc = c/bCase 2: c≤x≤bThe cumulative distribution function of X from c to b is given by:F(x) = ∫f(x)dx = ∫(2(b-x)/b(b-c))dx (evaluated from c to x)= (2(b-x)(x-c))/(b(b-c))evaluated from c to b= 1- (2(b-c)²)/(2b(b-c))= 1 - (b-c)/b= c/bTherefore, the cumulative distribution function of X is:F(x) = {0, if x < 0;c/b, if 0 ≤ x ≤ c;1 - (b-c)/b, if c ≤ x ≤ b;0, if x > b}b) E[X]:To find the expected value of X, we must find the mean of the distribution. For 0≤x≤c, the expected value isE[X] = ∫x.f(x)dx = ∫x(2x/bc)dx (evaluated from 0 to c)= c²/bFor c≤x≤b,E[X] = ∫x.f(x)dx = ∫x(2(b-x)/b(b-c))dx (evaluated from c to b)= (b+c)/3Therefore, the expected value of X isE[X] = (c²+ b²+ c.b)/(3b) = (b+ c/3)²c) Var(X):To calculate the variance of X, we must first calculate the expected value squared of X. E[X²]For 0≤x≤c,E[X²] = ∫x².f(x)dx = ∫x²(2x/bc)dx (evaluated from 0 to c)= c³/3bFor c≤x≤b,E[X²] = ∫x².f(x)dx = ∫x²(2(b-x)/b(b-c))dx (evaluated from c to b)= (2c² + b²)/3Therefore, the expected value squared of X isE[X²] = (c³/3b) + (2c² + b²)/3Now we can calculate the variance using the formula:Var(X) = E[X²] - [E[X]]²= [(c³/3b) + (2c² + b²)/3] - [(b+ c/3)²]= (c² - bc + b²)/9Therefore, the variance of X isVar(X) = (c² - bc + b²)/9d) Range of X:The range of X is [0,b], as given in the density function of X. The range of a continuous random variable is the set of values that the variable can take on. The range of X is from 0 to b because the function f(x) is only defined on this interval. Therefore, X cannot take on any values outside this range.

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Convergent

The given integral is ∫0 to [infinity] e^−7x dx. Determine whether the integral is convergent or divergent. If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.)Given integral is ∫0 to [infinity] e^−7x dx. To check the convergence or divergence of the given integral, we can use the following formula;∫a to [infinity] e^−x dx = e^−aGiven integral ∫0 to [infinity] e^−7x dx can be written as;∫0 to [infinity] e^−7x dx = e^−0...[as lower limit is zero]∫0 to [infinity] e^−7x dx = 1/7 [1/e^0 - 1/e^∞]∫0 to [infinity] e^−7x dx = 1/7 [1 - 0]∫0 to [infinity] e^−7x dx = 1/7 [1] = 1/7Since the integral is a finite value, the given integral is Convergent.The answer is Convergent.

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The expression for the ratio will be AB/BC = sin(∠BAC)/sin(∠CAB).

What is Expression?

An expression is mathematical phrase that can contain numbers, variables, operators, and functions, but does not have equal sign. It represents  value or  mathematical operation that can be evaluated or simplified.

we can use the Pythagorean theorem to find the distances between the houses, and then use trigonometric ratios to find the ratios between them.

Let's assume that Amy's house is located at point A, Barry's house is located at point B, and Chris's house is located at point C, as shown in the diagram below.

 C

 |

 |

---A--------

 |

 |

 B

a. To find the ratio of Barry's distance from Amy to Chris' distance from Amy, we can use the following trigonometric expressions:

tan(∠BAC) = AB/AC     (opposite/adjacent)

tan(∠CAB) = AC/AB     (opposite/adjacent)

Therefore,

AB/AC = 1/tan(∠BAC)

AC/AB = 1/tan(∠CAB)

b. To find the ratio of Barry's distance from Chris to Amy's distance from Chris, we can use the following trigonometric expressions:

tan(∠ACB) = AB/BC     (opposite/adjacent)

tan(∠ABC) = AC/BC     (opposite/adjacent)

Therefore,

AB/BC = tan(∠ACB)

AC/BC = tan(∠ABC)

c. To find the ratio of Amy's distance from Barry to Chris' distance from Barry, we can use the following trigonometric expression:

sin(∠BAC) = AB/BC     (opposite/hypotenuse)

sin(∠CAB) = AC/BC     (opposite/hypotenuse)

Therefore,

AB/BC = sin(∠BAC)

AC/BC = sin(∠CAB)

So,

AB/BC = sin(∠BAC)/sin(∠CAB).

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The probability of a bone density score greater than 0.18 is given as follows:

0.4286 = 42.86%.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation for this problem are given as follows:

[tex]\mu = 0, \sigma = 1[/tex]

The probability of a score greater than 0.18 is one subtracted by the p-value of Z when X = 0.18, hence:

Z = (0.18 - 0)/1

Z = 0.18

Z = 0.18 has a p-value of 0.5714.

Hence:

1 - 0.5714 = 0.4286 = 42.86%.

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The my supervisor cares about my opinions.

Scale Items Work Engagement Items – measured on a 1-7 scale (1-Never, 7-Always):  At my work, I feel bursting with energy; at my job, I feel strong and vigorous; when I get up in the morning I feel like going to work; I am enthusiastic about my job; I am proud of the work I do; I feel happy when I am working intensely; I am immersed in my work; I get carried away when I am working.

Job Tension Items – measured on 1-7 scale (1-Strongly disagree, 7-Strongly agree): My job is extremely stressful; very few stressful things happen to me at work; I feel a great deal of stress because of my job; I almost never feel stressed because of my work.

Career Satisfaction Items – measured on 1-7 scale (1-strongly disagree, 7-strongly agree): I am satisfied with the success I have achieved in my career; success I have made toward meeting my overall career goals; progress I have made toward meeting my goals for income; progress I have made toward meeting my goals for advancement; progress I have made toward meeting my goals for developing new skills.

Perceived Internal Mobility Items – measured on 1-5 scale (1-strongly disagree, 5-strongly agree): My organization views me as an asset to the organization; given my skills and experience, my organization views me as a value-added resource; there are many opportunities available for me in my organization.

Perceived Supervisory Support Items – measured on 1-7 scale (1-Strongly disagree, 7- Strongly agree): My supervisor strongly considers my goals and values; help is available from my supervisor when I have a problem; my supervisor really cares about my well-being; my supervisor would forgive an honest mistake on my part; my supervisor is willing to help me when I need a special favor; if given the opportunity, my supervisor would take advantage of me; my supervisor shows very little concern for me; my supervisor cares about my opinions.

Answer: The question asks about scale items measuring Work Engagement, Job Tension, Career Satisfaction, Perceived Internal Mobility, and Perceived Supervisory Support, all measured on different scales. Work Engagement is measured on a 1-7 scale (1-Never, 7-Always), Job Tension is measured on a 1-7 scale (1-Strongly disagree, 7-Strongly agree), Career Satisfaction is measured on a 1-7 scale (1-strongly disagree, 7-strongly agree), Perceived Internal Mobility is measured on a 1-5 scale (1-strongly disagree, 5-strongly agree), and Perceived Supervisory Support is measured on a 1-7 scale (1-Strongly disagree, 7- Strongly agree). In terms of Work Engagement, the items are: At my work, I feel bursting with energy; at my job, I feel strong and vigorous; when I get up in the morning I feel like going to work; I am enthusiastic about my job; I am proud of the work I do; I feel happy when I am working intensely; I am immersed in my work; I get carried away when I am working. For Job Tension, the items are: My job is extremely stressful; very few stressful things happen to me at work; I feel a great deal of stress because of my job; I almost never feel stressed because of my work. For Career Satisfaction, the items are: I am satisfied with the success I have achieved in my career; success I have made toward meeting my overall career goals; progress I have made toward meeting my goals for income; progress I have made toward meeting my goals for advancement; progress I have made toward meeting my goals for developing new skills. For Perceived Internal Mobility, the items are: My organization views me as an asset to the organization; given my skills and experience, my organization views me as a value-added resource; there are many opportunities available for me in my organization. Finally, for Perceived Supervisory Support, the items are: My supervisor strongly considers my goals and values; help is available from my supervisor when I have a problem; my supervisor really cares about my well-being; my supervisor would forgive an honest mistake on my part; my supervisor is willing to help me when I need a special favor; if given the opportunity, my supervisor would take advantage of me; my supervisor shows very little concern for me; my supervisor cares about my opinions.

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Charles will need to buy 10 bags of fertilizer to cover the entire garden.

What is Algebraic expression ?

In mathematics, an algebraic expression is a combination of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division, that represents a quantity or a relationship between quantities.

The area of the rectangular garden is:

A = length x width

A = 5 x 12

A = 60

To cover the entire garden, Charles needs:

Number of bags = (Area of garden) ÷ (Area covered by one bag)

Number of bags = 60  ÷ 6

Number of bags = 10 bags

Therefore, Charles will need to buy 10 bags of fertilizer to cover the entire garden.

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Answer:

Since the student is equally likely to select pizza, nachos, or chicken for lunch, the probability of selecting each of these options is 1/3.

The probability that the student does not select chicken is the probability of selecting pizza or nachos, which are the two options other than chicken. Since these two options are equally likely, the probability of selecting pizza or nachos is 1/2.

Therefore, the probability that the student does not select chicken is 1/2.

Jane sees 5 pandas in a nature magazine. Jane sees 7 fewer pandas than Quinn. Quinn sees 12 pandas.

Let's assume that Quinn sees x pandas.

We are told that Jane sees 7 fewer pandas than Quinn. Therefore, Jane sees x - 7 pandas.

We know that Jane sees 5 pandas. So we can set up an equation:

x - 7 = 5

To solve for x, we add 7 to both sides of the equation:

x - 7 + 7 = 5 + 7

x = 12

Therefore, Quinn sees 12 pandas.

We can check our answer by verifying that Jane sees 7 fewer pandas than Quinn:

12 - 7 = 5

The answer is consistent with the information given in the problem, and we can conclude that Quinn sees 12 pandas.

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The values of m and n for the polynomial [tex]p(x) = 3x^3 + 12x^2 - 15x - 3,[/tex] satisfying the given conditions, are m = 3 and n = -15.

To determine the values of m and n in the polynomial[tex]p(x) = mx^3 + 12x^2 + nx - 3[/tex], use the Remainder Theorem. According to the theorem, if a polynomial p(x) is divided by (x - a), the remainder is equal to p(a).

Given that when dividing p(x) by (x + 3) is zero, substitute -3 for x in the polynomial and set it equal to zero:

[tex]m(-3)^3 + 12(-3)^2 + n(-3) - 3 = 0[/tex]

Simplifying this equation gives us:

-27m + 108 + (-3n) - 3 = 0

-27m - 3n + 105 = 0

Next, we are given that the remainder when dividing p(x) by (x - 2) is 85. Using the same logic, we substitute 2 for x in the polynomial and set it equal to 85:

[tex]m(2)^3 + 12(2)^2 + n(2) - 3 = 85[/tex]

Simplifying this equation gives us:

8m + 48 + 2n - 3 = 85

8m + 2n + 45 = 85

Now we have a system of two equations with two variables:

-27m - 3n+ 105 = 0

8m + 2n + 45 = 85

Solving this system of equations will give us the values of m and n. By solving these equations, we find that m = 3 and n = -15.

Therefore, the values of m and n for the polynomial [tex]p(x) = 3x^3 + 12x^2 - 15x - 3,[/tex] satisfying the given conditions, are m = 3 and n = -15.

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Answer:

a. -1

b. 1

c. 6

Step-by-step explanation: f(any value) just means plug in the value given inside of the f and see what y value comes out. for the first on, go to where four it on the x-axis and you will see that the y-value is -1. Repeat the same thing for f(0) where you can look directly on the y-axis and get 1 as the answer. Lastly, look where -4 meets the y-axis and it will be 6.

11. The triangles are similar to each other by SAS rule of congruency.

12. The triangles are similar to each other by ASA rule of congruency.

13. The triangles are similar to each other by SSS rule of congruency.

Triangles that resemble one another but may not be precisely the same size are said to be comparable triangles. If two objects are similar in shape but have different sizes, they are considered to be comparable.

This indicates that comparable shapes superimpose one another when amplified or demagnified. The term "Similarity" refers to this characteristic of like shapes.

Now in the 1st given figure,

One pair of angles are equal to each other, and the sides of the triangle ACE and BCD are in proportion to each other.

So, we can say that the triangles are similar to each other.

In the 2nd figure,

One pair of angles are equal to each. Another pair is equal to each other as the lines are dissecting each other.

And the side is in equal proportion to each other. So, the triangles are congruent to each other.

In the last figure,

One side of the triangle are parallel to each other.

The other sides are in proportion to each other.

Hence, all the three cases triangles are similar to each other.

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