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Find m∠X.

Write your answer as an integer or as a decimal rounded to the nearest tenth.

m∠X = °

Part a

Angle ABC = angle CDA (given by the angle markers)

Angle BAC = angle DCA (alternate interior angles)

Segment AC = segment AC (reflexive property)

Through AAS (angle angle side) we can prove the two triangles are congruent. We have a pair of congruent angles, and we have a pair of congruent sides that are not between the previously mentioned angles.

If two triangles are congruent, they are always similar as well (scale factor = 1).

The same cannot be said the other way around. Not all similar triangles are congruent.

======================================================

Part b

Angle FGH = angle JIH (both shown to be 50 degrees)

Angle FHG = angle JHI (vertical angles)

We have enough information to prove the triangles to be similar triangles. This is through the AA (angle angle) similarity rule. Since FG and JI are different lengths, this means the triangles are not congruent.

======================================================

Part c

For each right triangle shown, divide the longer leg over the shorter leg

larger triangle: (long leg)/(short leg) = 6/3 = 2

smaller triangle: (long leg)/(short leg) = 3/2 = 1.5

The two results are different, so the sides are not in proportion to one another, therefore the triangles are not similar.

Any triangles that are not similar will also never be congruent.

======================================================

Part d

Use the pythagorean theorem to find that PQ = 5 and KL = 12

We have two triangles with corresponding sides that are the same length

So we use the SSS (side side side) triangle congruence theorem to prove the triangles congruent. The triangles are also similar triangles (scale factor = 1)

======================================================

In Mathematics, triangles can be congruent, similar, or neither. Congruency means the triangles have the same three sides and angles. Similarity means the triangles have the same shape but not necessarily the same size.

In Mathematics, particularly in Geometry, determining whether two triangles are congruent, similar, or neither is a pivotal concept. Triangles are congruent when they have exactly the same three sides and exactly the same three angles. On the other hand, triangles are similar when they have the same shape but not necessarily the same size.

To determine if triangles are congruent, you can use several postulates, including the Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Side-Angle (ASA) postulates. For triangle similarity, the Angle-Angle (AA) postulate is often used. In the absence of sufficient information, the triangles cannot be declared similar or congruent.

A flowchart to justify the congruence or similarity would begin by assessing if all corresponding angles and sides match. If so, the triangles are congruent. If only the angles match and the sides are proportional, then the triangles are similar. In the absence of either, the triangles are neither similar nor congruent.

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Answer: if he drives the car each month, he would spend $150 lesser than when he drives the truck.

Step-by-step explanation:

The car goes 25 miles on 1 gallon of gas. Jamaica drives 1000 miles per month, it means that the number of gallons of gas that he would use in a month is

1000/25 = 40 gallons of gas

If gasoline costs $2.50 per gallon and Jamaica chooses to buy a car, the cost of gas per month would be

2.5 × 40 = $100

The truck goes 10 miles on 1 gallon of gas. Jamaica drives 1000 miles per month, it means that the number of gallons of gas that he would use in a month is

1000/10 = 100 gallons of gas

If gasoline costs $2.50 per gallon and Jamaica chooses to buy a truck, the cost of gas per month would be

2.5 × 100 = $250

The difference between both costs is

250 - 100 = $150

Answer:

View Image

Step-by-step explanation:

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Answer:

B. 22

Step-by-step explanation:

124.06 = 2.35h + 72.36

124.06 - 72.36 = 2.35h

51.7 = 2.35h

51.7/2.35 = h

22 = h

Answer: it will take 2 days and the customer will pay $49

Step-by-step explanation:

Let x represent the number of days for which the cost would be the same.

At Joseph rentals, customers will pay $47 to rent a midsize car for the first day plus two dollars for each additional day. This means that the total cost of using Joseph rental for x days would be

47 + 2(x - 1) = 47 + 2x - 2

= 45 + 2x

At Fox rental, the price for a midsize car is $36 for the first day and $13 for every additional day beyond that. This means that the total cost of using Fox rental for x days would be

36 + 13(x - 1) = 36 + 13x - 13

= 23 + 13x

At the point where renting at either companies will cost the customer the same amount, then

45 + 2x = 23 + 13x

13x - 2x = 45 - 23

11x = 22

x = 22/11 = 2

The amount that the customer will psy is

23 + 13 × 2 = 49

Answer:

Step-by-step explanation:

given that a  bacteria culture initially contains cells and grows at a rate proportional to its size.

If P be the size then growth rate

[tex]P'=kP[/tex] where k is constant of proportionality

separate the variables as

[tex]\frac{dP}{P} =kdt\\ln P =kt+C\\P = Ae^{kt}[/tex]

If after 1 hour population is B (say)

[tex]B=Ae^{k} \\\\k = ln B - ln A[/tex]

then k = ln B - ln A

Using this

P(t) = [tex]Ae^{(lnB-lnA)t}[/tex]

b) P(e) = [tex]Ae^{(lnB-lnA)3}[/tex]

c) Rate of growth = [tex](ln B- ln A)Ae^{(lnB-lnA)3}[/tex]

Unless you give B value, d cannot be solved

Answer:

Step-by-step explanation:

Let x represent the maximum number of miles that she can drive.

Riley needs to rent a car while on vacation. The rental company charges $18.95, plus 16 cents for each mile driven. Converting 16 cents to dollars, it becomes 16/100 = $0.16

Assuming Riley drives the car for x miles, the total charge would be

0.16x + 18.95

If Riley only has $50 to spend on the car rental, it means that

0.16x + 18.95 = 50

0.16x = 50 - 18.95

0.16x = 31.05

x = 31.05/0.16 = 194.0625

The maximum number of miles that

she can drive is 194 miles.

Answer:

$251,618 is the answer

Step-by-step explanation:

From the previous question, we know he pays $698.94 monthly.

He has to make 360 payments. $698.94 * 360 = $251,618

Answer:

  A.  -2

  B.  -10

Step-by-step explanation:

The slope of a perpendicular line will be the negative reciprocal of the slope of the given line:

  -1/(1/2) = -2 . . . . slope of the perpendicular line

__

The y-intercept will let the given point satisfy the equation ...

  y = -2x +b

  2 = -2(-6) +b

  -10 = b . . . . . . . subtract 12. This is the y-intercept.

_____

The graph shows the two lines and the points they go through.

Answer:

  2 .The slope of Function A is less than the slope of Function B

Step-by-step explanation:

A graph of Function A shows it has a y-intercept of 4, the same as that of Function B. (Statements 3 and 4 are not correct.)

The slope of Function A is 2, which is less than the slope of 3 that Function B has. (Statement 2 is correct; statement 1 is not.)

_____

More detailed working

The slope of Function A can be figured easily between the points with x-values that differ by 1:

  m = (y3 -y2)/(x3 -x2) = (24-22)/(10-9) = 2/1 = 2 . . . . . Fun A has slope of 2.

The slope of Function B is the coefficient of x in the equation: 3.

__

The y-intercept of Function A can be found starting with point-slope form:

  y -22 = 2(x -9)

  y = 2x -18 +22

  y = 2x +4 . . . . . . . slope-intercept form

The intercept of +4 is the same as that of Function B.

The solution of the expression of the inequality - 6 ≥ 10 - 8x for x

would be;

⇒ x ≥ 2

What is Mathematical expression?

The combination of numbers and variables by using operations addition, subtraction, multiplication and division is called Mathematical expression.

Given that;

The expression of the inequality is;

⇒ - 6 ≥ 10 - 8x

Now,

Solve the inequality for x as;

The inequality is;

⇒ - 6 ≥ 10 - 8x

Add 8x both side, we get;

⇒ - 6 + 8x ≥ 10 - 8x + 8x

⇒ - 6 + 8x ≥ 10

Add 6 both side, we get;

⇒ - 6 + 8x + 6 ≥ 10 + 6

⇒ 8x ≥ 16

Divide by 8 both side, we get;

⇒ x ≥ 16/8

⇒ x ≥ 2

Hence, - 6 ≥ 10 - 8x ⇒  x ≥ 2

Thus, The solution of the expression of the inequality - 6 ≥ 10 - 8x, for x will be;

⇒ x ≥ 2

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Answer:

Correct statement: a. x₁ + x₂ + x₃ > 2

Step-by-step explanation:

The variables x₁, x₂ and x₃ takes value 0 if the projects are not done and 1 if the projects are done.

Consider that at least two projects are done, i.e. 2 or more projects are done.

This can happen in:

x₁ = 0, x₂ = 1 and x₃ = 1

x₁ = 1, x₂ = 0 and x₃ = 1

x₁ = 1, x₂ = 1 and x₃ = 0

x₁ = 1, x₂ = 1 and x₃ = 1

The statement (x₁ + x₂ + x₃ > 2) will be true only when all the variables takes the value 1.

This statement implies that 2 projects are definitely done.

Thus, the correct statement is (a).

Final answer:

Isaac has 134 square feet of the wall left to paint after subtracting the area he has already painted (28 square feet) from the total area of the wall (162 square feet).

Explanation:

The student's question is regarding an area calculation problem. Isaac is painting a wall with dimensions of 9 feet by 18 feet and has painted a 4 feet by 7 feet section so far. To find the area left to paint, we need to calculate the total area of the wall and subtract the area that's already been painted.

Step 1: Calculate the total area of the wall

The total area of the wall is:

(Length of the wall) × (Width of the wall) = 9 ft × 18 ft = 162 square feet.

Step 2: Calculate the area that has been painted

The area that Isaac has painted is:

(Length of painted section) × (Width of painted section) = 4 ft × 7 ft = 28 square feet.

Step 3: Calculate the area left to paint

To find the remaining area to paint:

(Total area of the wall) - (Area painted) = 162 sq ft - 28 sq ft = 134 square feet.

So, Isaac has 134 square feet of the wall left to paint.

Answer:

45%

Step-by-step explanation:

For simplicity, let use assume there are 100 students in the school.

No. of students to complete college = (30/100) x 100 = 30 Students

President wants to increase by 50% = (50/100) x 30 = 15 Students

New set goal = 30 + 15 = 45 students.

Total number of students = 100 students

Therefore;

Rate goal % = (45/100) x 100% = 45%

Answer:

Step-by-step explanation:

The fraction of tickets that are adult tickets is ...

  ((average price per ticket) - (child's ticket cost)) / (difference in ticket costs)

so the fraction of adult tickets is ...

  ((2881/548) -3.50)/(6.50 -3.50) = 321/548

Then the number of adult tickets is ...

  (321/548)·548 = 321

and the number of child tickets is ...

  548 -321 = 227

321 adult and 227 child tickets were sold that night.

_____

If you want to write an equation, you can let "a" represent the number of adult tickets sold. Total revenue is ...

  6.50a +3.50(548 -a) = 2881

  3.00a +1918 = 2881 . . . . . . eliminate parentheses

  3a = 963 . . . . . . . . . . . . . . . subtract 1918

  a = 321 . . . . . . . . . . . . . . . . . divide by 3

The number of child tickets is ...

  548 -a = 548 -321 = 227

Answer:

The temperature would be -5 degrees Fahrenheit

Step-by-step explanation: It's represented by a negative integer because 15 - 20 = -5. This means the temperature outside would be -5 degrees Fahrenheit.

Hope this helps! (:

The temperature would be -5 degrees Fahrenheit if The temperature outside is 15 degrees Fahrenheit . If the temperature drops 20 degrees,

It is defined as the conversion from one quantity unit to another quantity unit followed by the process of division, multiplication by a conversion factor.

It is given that:

The temperature outside is 15 degrees Fahrenheit . If the temperature drops 20 degrees

=15 - 20

= -5.

A negative sign means the temperature outside would be -5 degrees Fahrenheit.

Thus, the temperature would be -5 degrees Fahrenheit if The temperature outside is 15 degrees Fahrenheit. If the temperature drops 20 degrees,

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Answer:

Helen: 60mph and Anne: 50mph

Step-by-step explanation:

3.2r+2.8(r-10)=332 is the equation that we use, given the information we have.

We distribute and combine like terms and add 28 to both sides and divide by 6.

3.2r+2.8(r-10)=332

3.2r+2.8r-28=332

                6r=360

             6r/6=360/6

                  r=60  So, Helen's speed is 60mph.

Next, we'll solve Anne's speed.

r-10=50

60-10=50  So, Anne's speed is 50mph.

Anne's average speed was approximately 45 mph, and Helen's average speed was approximately 55 mph.

Let's denote Helen's average speed as "H" and Anne's average speed as "A." We are given that Helen drove for 3.2 hours, and Anne drove for 2.8 hours. We also know that Helen's average speed was 10 miles per hour faster than Anne's, so we can write this relationship as:

H = A + 10

Now, using the formula Speed = Distance / Time, we can express the distances traveled by Helen and Anne:

Distance covered by Helen = H * 3.2

Distance covered by Anne = A * 2.8

Given that the sum of their distances equals the distance between their homes (332 miles):

H * 3.2 + A * 2.8 = 332

Substituting the relationship H = A + 10, we get:

(A + 10) * 3.2 + A * 2.8 = 332

Solving this equation will provide us with Anne's average speed (A), and subsequently, we can find Helen's average speed (H) using the relationship H = A + 10.

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Answer:

2.78hrs

Step-by-step explanation:

Volume of water in the pool =πr2h

V = 3.142 * 2.5² *1.7

V = 33.38m³

Emptying the pool out at 12m³ per hour

= 33.38/12

= 2.78hrs

The constant amount of depreciation in the value of boat per year is $ 500

Solution:

When he bought the boat in 2004 he paid $26,500

Therefore,

Initial value in 2004 = $ 26500

In the year 2011, Ryan's boat had a value of $23,000

Value in 2011 = $ 23000

The value of the boat depreciated linearly

If the boat depreciation is linear, then the amount by which the value of boat depreciates must be constant.

Let x be the constant depreciation in the value of boat per year

Then we can say,

Value in 2011 = Initial value in 2004 - nx

Here, "n" is the number of years

2011 - 2004 = 7 years

Therefore,

23000 = 26500 - 7x

7x = 26500 - 23000

7x = 3500

Divide both sides by 7

x = 500

Thus the rate of depreciation per year is $ 500

The annual rate of change of the boat's value is approximately -71.43 dollars.

The annual rate of change of the boat's value can be calculated using the formula for slope of a line. We subtract the initial value from the final value and divide it by the number of years the boat has depreciated. In this case, the initial value is $26,500 and the final value is $23,000. The number of years is 7 (2011 - 2004). So the annual rate of change is ($23,000 - $26,500)/7 = -$500/7 = -71.43. Therefore, the annual rate of change of the boat's value is approximately -71.43 dollars.

The student is asking about the annual rate of change in the value of a boat, which is a problem related to linear depreciation. To solve this, we need to calculate the total amount the boat depreciated over a certain period and then divide by the number of years to get the annual rate.

Ryan's boat was worth $23,000 in 2011 and was purchased for $26,500 in 2004. The total depreciation over these 7 years is $26,500 - $23,000 = $3,500. To find the annual depreciation rate, we divide the total depreciation by the number of years: $3,500 ÷ 7 years = $500 per year.

Therefore, the annual rate of change of the boat's value is $500 per year, which means the boat's value decreased by $500 every year on average.

Answer:

  about 29.7 minutes

Step-by-step explanation:

If it take c minutes for Charles to mow the lawn by himself, it takes c+16 minutes for Peter. The two of them working together can mow in one minute this fraction of the entire lawn:

  1/c + 1/(c+16) = 1/18

Multiplying by 18c(c+16), we get ...

  18(c +16) + 18(c) = c(c+16)/18

  36c +288 = c^2 +16c

  c^2 -20c = 288 . . . . . subtract 36c

  c^2 -20c +100 = 388 . . . . . add (20/2)^2 = 100 to complete the square

  (c -10)^2 = 388

  c = 10 +√388 ≈ 29.6977 . . . . . take the positive square root

It takes Charles about 29.7 minutes to mow the lawn by himself.

Answer:

206

Step-by-step explanation:

We have been given that Biologists tagged 103 fish in a lake January 1. On February 1, they returned and collected a random sample of 24 fish, 12 of which had been previously tagged.

To find the number of fish in the lake, we will use proportions because ratio of tagged fish and collected fish on February 1 will be equal to ratio of tagged fish and total fish on January 1.

[tex]\frac{\text{Tagged fish}}{\text{Collected fish}}=\frac{12}{24}[/tex]

Upon substituting the number of tagged fish in our proportion, we will get:

[tex]\frac{103}{\text{Total fish}}=\frac{12}{24}\\\\\frac{103}{\text{Total fish}}=\frac{1}{2}[/tex]

Cross multiply:

[tex]1\cdot \text{Total fish}=103\cdot 2\\\\\text{Total fish}=206[/tex]

Therefore, there are approximately 206 fishes in the lake.

Answer:

Option C - determine whether [tex]9-4(-x)^2[/tex] is equivalent to [tex]-(9-4x^2)[/tex] or not.

Step-by-step explanation:

To find : Which statement best describes how to determine whether [tex]f(x) = 9-4x^2[/tex] is an odd function?

Solution :

We have a property for odd functions,

Let f(x) be an odd function then it must satisfy

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

Now, we have been given the function [tex]f(x) = 9-4x^2[/tex]

For this function to be odd, it must satisfy the above property.

Replace x with -x,

[tex]f(-x)=9-4(-x)^2[/tex]

and

[tex]-f(x)=-(9-4x^2)[/tex]

Hence, in order to the given function to be an odd function, we must determine whether [tex]9-4(-x)^2[/tex] is equivalent to [tex]-(9-4x^2)[/tex] or not.

Therefore, C is the correct option.

Answer:

500000 people

Step-by-step explanation:

The population grew by 600,000 which is 120% the earlier prediction by the town council.

Using direct proportion

600,000  -------- 120%

X              --------- 100%

X = (600000 × 100) ÷ 120 = 500000

Therefore the earlier prediction by the town council is 500000 people

The student's question is a mathematical problem calculating population growth predictions. The town council of Grand Island originally predicted a growth of 500,000 people, which is 20% less than the actual growth of 600,000 people.

To determine the prediction made by the town council, we can use the fact that the actual growth exceeded the prediction by one fifth (or 20%). If the actual growth was 600,000 people, the predicted growth can be calculated by dividing 600,000 by 1.2, as the actual growth represents 120% of the predicted value (100% original prediction + 20% excess).

Calculating the Predicted Population Growth

To find the town council's predicted growth, we can set up the equation:

Therefore, the town council had originally predicted that the population of Grand Island, Nebraska, would increase by 500,000 people between 1995 and 2005.

Answer:

[tex]m\angle R=69.4^o[/tex]

Step-by-step explanation:

we know that

In the right triangle PQR

[tex]tan(R)=\frac{PQ}{QR}[/tex] ----> by TOA (opposite side divided by adjacent side)

substitute the given values

[tex]tan(R)=\frac{8}{3}[/tex]

using a calculator

[tex]m\angle R=tan^{-1}(\frac{8}{3})=69.4^o[/tex]

Answer:

The third option is correct i.e. 15 x + 30 y minus 220.

Step-by-step explanation:

We have to choose expression from the option that is equivalent to

[tex]30(\frac{1}{2}x - 2) + 40(\frac{3}{4}y - 4)[/tex]

Now, [tex]30(\frac{1}{2}x - 2) + 40(\frac{3}{4}y - 4)[/tex]

= 15x - 60 + 30y - 160

= 15x + 30y - 220

Therefore, the third option is correct i.e. 15 x + 30 y minus 220. (Answer)

Step-by-step explanation: C.

Answer:

d = $5.50 - p

Step-by-step explanation:

Answer: d = $5.50 - p

Step-by-step explanation:

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Answer:

36π cm^2.

Step-by-step explanation:

This is a sphere . Surface area =  4πr^2.

This sphere has surface area = 4π3^2

= 36π.

The surface area of the sphere would be = 36πcm². That is option C.

Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object.

here, we have,

to calculate the surface area of a sphere:

The surface area of a sphere can be calculated through the use of the formula = 4πr²

Where,

radius (r) = 3 cm

surface area

=4πr²

= 4π × 3²

= 36π cm² ( in the terms of π)

Hence, The surface area of the sphere would be = 36πcm². That is option C.

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Answer:

The average age was minimum at 1954 and the average age is 25.5.

Step-by-step explanation:

The given quadratic function is

[tex]f(x)=0.0039x^2-0.42x+36.79[/tex]

It models the​ median, or​ average, age,​ y, at which men were first married x years after 1900.

In the above equation leading coefficient is positive, so it is an upward parabola and vertex of an upward parabola, is point of minima.

We need to find the year in which the average age was at a minimum​.

If a quadratic polynomial is [tex]f(x)=ax^2+bx+c[/tex], then vertex is

[tex]Vertex=(-\dfrac{b}{2a},f(-\dfrac{b}{2a}))[/tex]

[tex]-\dfrac{b}{2a}=-\dfrac{(-0.42)}{2(0.0039)}=53.846153\approx 54[/tex]

54 years after 1900 is

[tex]1900+54=1954[/tex]

Substitute x=54 in the given function.

[tex]f(54)=0.0039(54)^2-0.42(54)+36.79=25.4824\approx 25.5[/tex]

Therefore, the average age was minimum at 1954 and the average age is 25.5.

The year when the average age at first marriage was at a minimum in a specific U.S. city was approximately 1954. The average age at first marriage for that year was approximately 28.4 years.

To find the year when the average age at first marriage was at a minimum, we need to determine the x-value at the vertex of the quadratic function. The x-value at the vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic function. For the given function f(x) = 0.0039x2 - 0.42x + 36.79, the x-value at the vertex is x = -(-0.42)/(2*0.0039) = 53.85. Since the x-value represents years after 1900, we add 1900 to get the year: 1900 + 53.85 ≈ 1954.

To find the average age at first marriage for that year, we substitute x = 53.85 into the quadratic function. f(53.85) = 0.0039(53.85)2 - 0.42(53.85) + 36.79 ≈ 28.4. Therefore, the average age at first marriage for the year 1954 was approximately 28.4 years.

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Step-by-step explanation:

[tex]\lim_{n \to \infty} \sum\limits_{k=1}^{n}f(x_{k}) \Delta x = \int\limits^a_b {f(x)} \, dx \\where\ \Delta x = \frac{b-a}{n} \ and\ x_{k}=a+\Delta x \times k[/tex]

In this case we have:

Δx = 3/n

b − a = 3

a = 1

b = 4

So the integral is:

∫₁⁴ √x dx

To evaluate the integral, we write the radical as an exponent.

∫₁⁴ x^½ dx

= ⅔ x^³/₂ + C |₁⁴

= (⅔ 4^³/₂ + C) − (⅔ 1^³/₂ + C)

= ⅔ (8) + C − ⅔ − C

= 14/3

If ∫₁⁴ f(x) dx = e⁴ − e, then:

∫₁⁴ (2f(x) − 1) dx

= 2 ∫₁⁴ f(x) dx − ∫₁⁴ dx

= 2 (e⁴ − e) − (x + C) |₁⁴

= 2e⁴ − 2e − 3

∫ sec²(x/k) dx

k ∫ 1/k sec²(x/k) dx

k tan(x/k) + C

Evaluating between x=0 and x=π/2:

k tan(π/(2k)) + C − (k tan(0) + C)

k tan(π/(2k))

Setting this equal to k:

k tan(π/(2k)) = k

tan(π/(2k)) = 1

π/(2k) = π/4

1/(2k) = 1/4

2k = 4

k = 2