Types of discontinuities

Identify and Classify the discontinuities in the function, please.

The shortest distance between point P and Q is 9.4.

The distance formula used in finding the distance between two points is expressed as;

D = √( ( x₂ - x₁ )² + ( y₂ - y₁ )² )

Given the data in the question;

Point P (-4,-2)

x₁ = -4

y₁ = -2

Point Q ( 4,3)

Plug the given values into the distance formula and simplify.

D = √( ( x₂ - x₁ )² + ( y₂ - y₁ )² )

D = √( ( 4 - (-4) )² + ( 3 - (-2) )² )

D = √( ( 4 + 4 )² + ( 3 + 2 )² )

D = √( 8² + 5² )

D = √( 64 + 25 )

D = √89

D = 9.4

Therefore, the distance is 9.4.

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The solution is,  the equation that represents Yulian’s situation, where x represents the time and y represents the amount of water is

y-29 = -7/2 (x-6).

Here, we have,

The given data is

time (h):    2   4   6   8

gallons:  43 36 29 22

A graph of the data shows a linear relationship.

Therefore in slope-intercept form, let

(y - 29)/(x - 6) = m

where m = the slope of the straight line.

Calculate the slope.

m = (43 - 22)/(2 - 8) = -7/2

Answer:  y-29 = -7/2 (x-6)

Hence, The solution is,  the equation that represents Yulian’s situation, where x represents the time and y represents the amount of water is

y-29 = -7/2 (x-6).

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Angle Y from the given triangle XYZ is 62° and sine of ∠Y is 60/68.

From the given triangle XYZ, XY=32 and YZ=68.

By using Pythagoras theorem, we get

YZ²=XY²+XZ²

68²=32²+XZ²

XZ²=68²-32²

XZ²=3600

XZ=√3600

XZ=60

We know that, sinθ= Opposite/Hypotenuse

sinY=XZ/YZ

sinY=60/68

sinY=0.8823

Y=61.92

Y≈62°

Therefore, angle Y from the given triangle XYZ is 62° and sine of ∠Y is 60/68.

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Answer: What are we finding?

Step-by-step explanation:

The polynomial function of least degree with integer coefficients is: P(x) = a(x³ + 14x² - 31x + 16)

Given that the properties of the polynomial are

If a polynomial has the given zeros 1 with a multiplicity of 2 and another zero at 4i, then its factors can be written as:

P(x) = a(x - 1)²(x - 4i)(x + 4i)

Multiplying out these factors, we get:

P(x) = a(x - 1)²(x + 16)

Expanding the right-hand side, we get:

P(x) = a(x³ + 14x² - 31x + 16)

Therefore, the polynomial function is: P(x) = a(x³ + 14x² - 31x + 16)

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Answer: The mixed numbers in ascending order are:

2 5/9 , 3 2/9 , 3 3/9 , 3 4/9 , 4 1/9

Step-by-step explanation:

To put these mixed numbers into ascending order, we need to convert all of them to improper fractions and then compare them.

3 4/9 = 31/9

4 1/9 = 37/9

3 3/9 = 30/9

3 2/9 = 29/9

2 5/9 = 23/9

so  in ascending order (from smallest to largest)

2 5/9 , 3 2/9 , 3 3/9 , 3 4/9 , 4 1/9

The sale prices of the four items are Coat: $111.99, Blouse: $35.19, Shoes: $71.99, Jeans: $26.39 , Using the discount percentage and Using the discount amount are two different ways to find the sale prices.

To find the sale price of each item, we need to multiply the regular price by the discount percentage, which is 20% or 0.2.

Then we subtract this discount amount from the regular price to get the sale price. We can use the following formula:

Sale price = Regular price - Discount amount

Using this formula, we can calculate the sale prices of the four items:

Coat: Sale price = $139.99 - ($139.99 x 0.2) = $111.99

Blouse: Sale price = $43.99 - ($43.99 x 0.2) = $35.19

Shoes: Sale price = $89.99 - ($89.99 x 0.2) = $71.99

Jeans: Sale price = $32.99 - ($32.99 x 0.2) = $26.39

Therefore, the sale prices of the four items are:

Coat: $111.99, Blouse: $35.19, Shoes: $71.99, Jeans: $26.39.

b. There are different ways to find the sale prices. Here are two methods:

Method 1: Using the discount percentage

Multiply the regular price of each item by the discount percentage (20% or 0.2).

Subtract the result from step 1 from the regular price to find the sale price.

Method 2: Using the discount amount

Multiply the regular price of each item by the discount percentage (20% or 0.2) to find the discount amount.

Subtract the discount amount from the regular price to find the sale price.

Both methods will give you the same sale prices, but some people may find one method easier or more intuitive than the other.

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One of the potential rational roots of 5x³-7x+11 is ±1, ±11, ±1/5, ±11/5. The correct answer is (B).

Rational Root Theorem states that if a polynomial has integer coefficients, then any rational root of the polynomial must have the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

For the given polynomial:

f(x) = 5x³-7x+11

constant term is 11

leading coefficient is 5.

Therefore, any potential rational roots of f(x) must have the form:

p/q

where p is a factor of 11 and q is a factor of 5.

The factors of 11 are ±1 and ±11, and the factors of 5 are ±1 and ±5. Therefore, the potential rational roots of f(x) are:

±1/1, ±11/1, ±1/5, ±11/5

which simplify to:

±1, ±11, ±1/5, ±11/5

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All the statements are true statements for a linear function

The list of options represent the given parameters in this question

Next, we test each of the options and state if they are true or not

So, we have

A. The rate of change is non-zero.

This is true because a linear function that has a zero rate is just a horizontal line

B. The domain is all real numbers.

This is true, otherwise when stated

C. The range is all real numbers.

This is true, otherwise when stated

D. There is one x-intercept.

This is true, because it intersects with the x-axis once

E. There is one y-intercept.

This is true, because it intersects with the y-axis once

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If the linear equation 2x+3y=14 is rewritten in function notation as f(y)=−32y+7, the dependent variable is x

In the linear equation 2x+3y=14, we can solve for y by isolating it on one side of the equation:

2x + 3y = 14

3y = -2x + 14

y = (-2/3)x + 14/3

Now, we can write this equation in function notation by replacing y with f(y) and x with the independent variable, usually denoted by 'x'. Therefore, we have:

f(y) = (-2/3)x + 14/3

The function f(y) tells us the value of x for any given value of y. In other words, y is the independent variable, and x is the dependent variable.

Therefore, the dependent variable in the function f(y) = (-2/3)x + 14/3 is x. The value of x is determined by the value of y, which is the input to the function. This means that for any value of y, we can determine the corresponding value of x by plugging in y into the equation and solving for x.

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At a $170 unit price, the producer's excess is $0.

To calculate the producer's surplus at a unit price of p = 170, we need to find the corresponding quantity supplied.

We can start by setting the supply equation equal to the unit price and solving for q:

p = 150 + 0.01q

170 = 150 + 0.01q

20 = 0.01q

q = 2000

So at a unit price of p = 170, the quantity supplied is q = 2000.

To find the producer's surplus, we need to calculate the cost of producing the 2000th unit, which is the marginal cost at this level of production. We can do this by taking the derivative of the supply equation with respect to q:

C(q) = 150q + 0.005q^2

MC(q) = dC/dq = 150 + 0.01q

At q = 2000, the marginal cost is:

MC(2000) = 150 + 0.01(2000) = 170

So the cost of producing the 2000th unit is $170.

The producer's surplus is the difference between the unit price and the marginal cost at the corresponding level of production:

PS = p - MC(q) = 170 - 170 = $0

Therefore, the producer's surplus at a unit price of $170 is $0.

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The overall probability of drawing B-R-A-D in that order is 1/24.

Option C is the correct answer.

We have,

The probability of drawing the letters B-R-A-D in that order is equal to the product of the probabilities of drawing each letter in that order.

Since we are drawing without replacement, the probability of drawing the first letter B is 1/4.

Then, the probability of drawing the second letter R is 1/3

(since one letter has already been drawn and removed from the hat).

Similarly, the probability of drawing the third letter A is 1/2 and the probability of drawing the fourth letter D is 1/1

(since all the other letters have already been drawn and removed).

The overall probability of drawing B-R-A-D in that order.

= (1/4) x (1/3) x (1/2) x (1/1)

= 1/24

Thus,

The overall probability of drawing B-R-A-D in that order is 1/24.

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Answer:

5(-6 + 2n) can be simplified using the distributive property of multiplication as follows:

5(-6 + 2n) = 5*(-6) + 5*(2n)

= -30 + 10n

Therefore, the simplified expression is -30 + 10n.

If  u = - 3i + 2i,  v = 2i - 2j and w = - 4j, 5u(3v - 4w) is a scalar quantity with a value of 440.

To evaluate the expression 5u(3v - 4w), we need to first perform the scalar multiplication and then the vector subtraction. We can start by computing the scalar multiplication of 3v - 4w:

3v - 4w = 3(2i - 2j) - 4(-4j) = 6i + 14j

Next, we can perform the scalar multiplication of u and the vector 6i + 14j:

5u(3v - 4w) = 5(-3i + 2j)(6i + 14j)

Expanding the product, we get:

5(-18i² + 36ij + 42ij - 28j²)

Since i² = j² = -1, we can simplify this expression as:

5(18 + 70) = 440

Therefore, 5u(3v - 4w) is a scalar quantity with a value of 440.

The key concept used in this problem is the distributive property of scalar multiplication over vector addition and subtraction. This property allows us to simplify expressions that involve scalar multiplication and vector operations by distributing the scalar to each component of the vector.

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Answer:

False

Step-by-step explanation:

This is because runner 1's median line falls on 5.6-5.7s while runner 2's falls on 6.4s

(Hopes this helps :) )

Answer:

Mackenzie would need to invest approximately $91,670, to the nearest ten dollars, for the value of the account to reach $194,000 in 11 years.

Step-by-step explanation:

We can use the formula for compound interest to solve the problem:

A = P(1 + r/n)^(nt)

where:

A = the amount of money in the account after t years

P = the principal amount (the initial investment)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the number of years

In this case, we want to find the principal amount P, given that A = $194,000, r = 6.5% = 0.065, n = 12 (compounded monthly), and t = 11 years. Plugging in these values, we get:

194000 = P(1 + 0.065/12)^(12*11)

Simplifying the right side, we get:

194000 = P(1.00541666667)^132

Dividing both sides by (1.00541666667)^132, we get:

P = 194000 / (1.00541666667)^132

P ≈ $91,670.

Therefore, Mackenzie would need to invest approximately $91,670, to the nearest ten dollars, for the value of the account to reach $194,000 in 11 years.

The answers for the conversion of the given masses in grams and kilograms are:

The conversion factor between kilogram and grams is given below:

The conversion of the given masses is as follows:

19 3/5 kg = 19 × 1000 + 3/5 × 1000

19 3/5 kg = 19,600 g

3.4 kg + 3/10 kg = 3,400 g + 300 g / 10

3.4 kg + 3/10 kg = 3,700 g

8.5 kg - 45 g = 8,500 g - 45 g

8.5 kg - 45 g  = 8,455 g

The mass of the dictionary is 1200 g = 1.2 kg

The difference in mass between the dictionary and the book is 1.2 kg - 0.4 kg = 0.8 kg

difference in mass = 800 g

1/2 kg + 50 g + 3.9 kg = 500 g + 50 g + 3,900 g

1/2 kg + 50 g + 3.9 kg = 4,450 g

3 kg 15 g = 3,015 g

1/4 kg = 250 g

3,015 g - 250 g = 2,765 g

7/10 kg + 7/100 kg = 700 g + 7 g

7/10 kg + 7/100 kg = 707 g

8 kg 8 g + 4 1/2 kg = 8,008 g + 4 × 1000 + 500 g

8 kg 8 g + 4 1/2 kg = 12,508 g

8 kg 8 g + 4 1/2 kg = 12.508 kg

Lisa's mass = 2/5 × 75 kg = 30 kg

Lisa's mass = 30,000 g

Mass of one box = 0.6 kg = 600 g

Mass of 12 boxes = 12 × 600 g

Mass of 12 boxes = 7,200 g

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The characteristics of the quadratic function and its graph are listed below

From the question, we have the following parameters that can be used in our computation:

The graph of the quadratic function

As a general rule

The domain of a quadratic function is the set of all real values

For the range, we have

The graph has a maximum value at y = -1

Also, the values of the function output (y values) decreases

This means that the range is y <= -1

For other key features, we have

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A transversal t intersects two parallel lines a and b, forming two groups of angles.

On top line a, starting from the top left, clockwise, angles are 1, 2, 3, and 4. On the below line b, starting from the top left, clockwise, angles are 5, 6, 7, and 8.

In the figure, angles 1 and 5 are corresponding angles, and both lines are intersected by transversal t. Complete the statements to prove that m∠1 = m∠5.

(given) m∠1 + m∠3 = 180° (Linear Pair Theorem)

m∠5 + m∠6 = 180° (Linear Pair Theorem)

m∠1 + m∠3 = m∠5 + m∠6 (Vertical Angles Theorem)

m∠3 = m∠6 (Subtraction Property of Equality)

m∠1 = m∠5 (Subtraction Property of Equality)

Thus, this can be concluded regarding the given scenario.

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The equation of the line in slope-intercept form is y = -2x - 3.

In this case, we know that the line passes through the point (-6,9) and has a slope of -2. Therefore, we can use the point-slope form of the equation to find the equation in slope-intercept form.

Point-slope form:

y - y₁ = m(x - x₁)

Substituting the values we know, we get:

y - 9 = -2(x - (-6))

y = -2x - 3

Therefore, the equation of the line in slope-intercept form is y = -2x - 3.

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The ratio of boys to girls in Mrs. Franco's class is 3:2 .

Here, we have,

The Ratio is defined as the comparison of two quantities that have the same units .

In the question ,

it is given that

In Mrs. Franco's class

For every 6 boys there are 4 girls in the class

we have to find the ratio of , boys to girls

the number of boys = 6

the number of girls = 4

So , the ratio can be written as

boys / girls = 6/4

writing the ratio in the simplest form , we get

boys/girls = 3/2

the ratio is 3:2   .

Therefore , The ratio of boys to girls in Mrs. Franco's class is 3:2 .

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complete question:

In Mrs. Franco‘s class for every 64 is there a April right the ratio of boys to girls in simplest form

The volume of the solid as shown in the diagram is 756 ft³.

Volume is the space occupied by a solid.

To caculate the volume of the solid, we use the formula below

Note: The solid is made up of a prism and a pyramid.

Formula:

Where:

From the diagram,

Given:

Substitute these values into equation 1

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By definitions of supplementary angles and by algebra properties, the value of the angle is equal to 118°.

In this problem we get a geometric system formed by a line cutting two parallel lines, from which we find two angles that are supplementary, that is, two angles whose sum equals 180°. Thus, the expression is described below:

(3 · n - 47) + (n + 7) = 180°

Now we proceed to determine to clear variable n by algebra properties:

4 · n - 40 = 180

4 · n = 220

n = 55

And the value of angle ABC is equal to:

m ∠ ABC = 3 · n - 47

m ∠ ABC = 3 · 55 - 47

m ∠ ABC = 118°

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The volume of one salt lick is 125/1728 feet which is less than one cubic foot.

All the edges of the cube are congruent with each other. Suppose that: The side length of the considered cube is L units. Then, we get:

V = L³ cubic units.

The edge length of the cube is 5/12 feet. Then the volume of the cube will be given as,

V = (5/12)²

V = 125 / 1728 feet

The volume is less than one cubic foot.

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The total thickness of a stack that contains 1 dime and 1 penny would be 40 millimeters.

If one should find the combined thickness of a collection containing 1 dime and 1 penny, adding both coins' individual thicknesses together is a suitable solution. According to the listed table, a dime's measure is 1 millimeter, while a penny's is 1 7 / 20 millimeters. Consequently, the complete depth of this selection could be calculated as follows:

= Thickness of dime + thickness of penny

= 1 7 / 20 + 1 1 / 2

= 27 / 20 + 3 / 2

= 27 / 20 + 30 / 20

= 57 / 20

= 2 17 / 20 mm

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a) The relative rate of growth of the bacteria population is approximately 6,300%.

b) The initial size of the culture was approximately 2 bacteria.

c) The function that models the number of bacteria is n(t) = [tex]27e^{(0.88t)[/tex]

d) The number of bacteria after 4.5 hours is approximately 3,106.

e) It takes approximately 5.51 hours for the number of bacteria to reach 25,000.

(a) To find the relative rate of growth of the bacteria population, we can use the formula:

Relative rate of growth = (Final count - Initial count) / Initial count x 100%

Using the values given in the problem, we get:

Relative rate of growth = (12,800 - 200) / 200 x 100% ≈ 6,300%

(b) To find the initial size of the culture, we can use the formula:

Final count = Initial count x e^(relative rate of growth x time)

We can rearrange this formula to solve for the initial count:

Initial count = Final count / e^(relative rate of growth x time)

Using the values given in the problem and solving for the initial count, we get:

Initial count = 200 / [tex]e^{(6300\% \times 2)[/tex] ≈ 2

(c) The function that models the number of bacteria n(t) after t hours is:

n(t) = n₀[tex]e^{(rt)[/tex]

To find the values of n₀ and r, we can use the values given in the problem:

n(2) = 200 = n₀[tex]e^{(2r)[/tex]

n(6) = 12,800 = n₀[tex]e^{(6r)[/tex]

Dividing the second equation by the first equation, we get:

[tex]e^{(4r)[/tex] = 64

Taking the natural logarithm of both sides, we get:

4r = ln(64)

r = ln(64) / 4 ≈ 0.88

Substituting this value of r into the first equation and solving for n₀, we get:

200 = n₀[tex]e^{(0.88 \times 2)[/tex]

n₀ ≈ 27

Therefore, the function that models the number of bacteria is:

n(t) = [tex]27e^{(0.88t)[/tex]

(d) To find the number of bacteria after 4.5 hours, we can use the function we found in part (c):

n(4.5) = 27[tex]e^{(0.88 \times 4.5)[/tex] ≈ 3,106

(e) To find the number of hours it takes for the number of bacteria to reach 25,000, we can use the function we found in part (c):

25,000 = 27[tex]e^ {(0.88t)[/tex]

Dividing both sides by 27 and taking the natural logarithm of both sides, we get:

ln(925.93) = 0.88t

t ≈ 5.51

In summary, we used the formula for relative rate of growth to find the rate of growth of the bacteria population, and used this information to find the initial size of the culture using exponential growth functions.

We found the function that models the number of bacteria, and used it to find the number of bacteria after a given time and the time it takes for the number of bacteria to reach a certain value.

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The equation of the trend line in slope intercept form is: K = -2J + 28

The formula for the equation of a line in slope intercept form is expressed in the form:

y = mx + c

where:

m is slope

c is y-intercept

Now, from the given line of best fit, we see that:

At x = 0, y = 28

Thus, c = 28

To get the slope, we will take 2 coordinate points which are:

(2, 24) and (4, 20)

Thus:

m = (20 - 24)/(4 - 2)

m = -2

Thus, the equation is:

K = -2J + 28

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Answer:

25 miles

Step-by-step explanation:

Robert runs 7 1/5 miles on Saturday. To find out how much he runs on Monday and Wednesday, we can multiply 7 1/5 by 2:

7 1/5 x 2 = 14 2/5

Robert runs 14 2/5 miles on both Monday and Wednesday, so his total distance for the week is:

7 1/5 + 14 2/5 + 14 2/5 = 35 miles

Robert wants to run a total of 60 miles this week, so he needs to run:

60 - 35 = 25 miles

Therefore, Robert needs to run 25 more miles to reach his goal of running 60 miles this week.

Using the given information, the total distance travelled by the golf ball is 288 m

Now, From the question, we are to calculate the total distance travelled by the golf ball

The distance travelled by the golf ball can be calculated by using the formula,

Distance = Speed × Time

From the given information,

The golf ball rolls at a speed of 8 m/s for 12 seconds

Thus, The distance travelled at this time is

Distance = 8 m/s × 12 s

Distance = 96 m

Also, Mandy hits the golf ball and it rolls for 16 seconds at a speed of 12 m/s

The distance travelled at this time is

Distance = 12 m/s × 16 s

Distance = 192 m

So, The total distance travelled by the golf ball is 96 m + 192 m

= 288 meters

Hence, the total distance travelled by the golf ball is 288 m

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