Mai is jogging from her house to school. Her school is 2 5/6 miles from her house. She has gone 1 3/5 miles so far. How many miles does Mai still have to jog? Write your answer as a mixed number in simplest form.

To produce 60 liters of a 55% acid solution, we would need 40 liters of the 50% acid solution and 20 liters of the 65% acid solution.

Let number of liters of 50% acid solution needed be = "x", and

Let number of liters of 65% acid solution needed to produce 60 liters of a 55% acid solution. be = y,

To solve for x and y, we can use the following system of equations:

The "total-volume" of the solution is 60 liters,

⇒ x + y = 60,    ...equation(1)

We know that, the total amount of acid in the solution is equal to the concentration of the final solution times its volume;

⇒ 0.50x + 0.65y = 0.55(60)     ...equation(2),

On simplifying equation(2),

We get,

⇒ 0.50x + 0.65y = 33,

From equation(1), we have ⇒ x = 60 - y,

Now, we substitute "x" into the second equation and solve for y)

⇒ 0.50(60 - y) + 0.65y = 33,

⇒ 30 - 0.50y + 0.65y = 33,

⇒ 0.15y = 3,

⇒ y = 20

So, we need 20 liters of the 65% acid solution.

⇒ x + y = 60,

⇒ x + 20 = 60,

⇒ x = 40,

Therefore, we need 40 liters of the 50% acid solution and 20 liters of the 65% acid solution to produce 60 liters of a 55% acid solution.

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The volume of the cylinder in terms of b is expressed as: (20b³ + 12b²)π cubic units.

Where r represents the radius of a cylinder and h represents its height, the volume of the cylinder can be calculated using the formula:

V = πr²h.

Given the following:

Radius (r) = 2b

Height (h) = 5b + 3

Volume (V) = π * (2b)² * (5b + 3)

Volume (V) = π * 4b² * (5b + 3)

Volume (V) = π * 20b³ + 12b²

Volume (V) = (20b³ + 12b²)π cubic units

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The area under the curve is 522 units².

The area under the curve is calculated by integrating the function and finding the definite integra as shown below.

f(x) = 8x + 10

The integra of the function;

f(x)' = 4x² + 10x

Find the definite integra at the boundaries, x = 19 and x = 22

f(19)' = 4(19)² + 10(19)

F(19)' = 1,634

f(22)' = 4(22)² + 10(22)

F(22)' = 2,156

Area = 2,156 - 1,634 = 522 units²

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The value of h is ( -1/3, -2/3)

Quadratic equation is a type of equation in which the highest power of variable is 2. A quadratic equation is represented as ax²+bx + c = 0

Solving, 9h²+9h = -2

9h²+9h+2 = 0

9h² +6h +3h +2 = 0

(9h²+6h) ( 3h +2) = 0

3h( 3h + 2)+1 ( 3h+2) = 0

(3h+2)(3h+1) = 0

Therefore ;

3h+2 = 0

3h = -2

h = -2/3 or

3h+1 = 0

3h = -1

h = -1/3

therefore the value of h is ( -1/3, -2/3)

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Answer:

1) Michele

2) f(t)

3) Elisa's puppy

4) g(x)

Step-by-step explanation:

This is because Michele's situation has a rate of change of -10, f(t) has -6.2, Elisa's puppy's situation has a rate of change of +5, and g(x) has 8.1

Hence,

-10 < -6.2 < 5 < 8.1

Hope this helps and be sure to mark this as brainliest! :)

if Dwayne made 10 hours of long-distance calls at a rate of $0.10 per minute, his bill for that month would be $60. However, if the rate is different, the bill will be different as well.

The cost of long-distance calls can vary depending on the service provider and the country being called. Without that information, it's difficult to give an accurate answer to this question.

Assuming a standard rate of $0.10 per minute for long-distance calls, we can calculate Dwayne's bill as follows:

10 hours = 600 minutes (since there are 60 minutes in an hour)

600 minutes x $0.10 per minute = $60

Therefore, if Dwayne made 10 hours of long-distance calls at a rate of $0.10 per minute, his bill for that month would be $60. However, if the rate is different, the bill will be different as well.

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Therefore, the general solution to the differential equation is

Y = e⁽ˣ⁾2+X+C) - 1 or Y = -e⁽ˣ⁾2+X+C) - 1

A differential equation is an equation that connects the derivatives of one or more unknown functions. It is an equation that uses the derivatives of a function or functions, in other words. Many physical processes, including the motion of objects under the influence of forces, the movement of fluids, and the spread of disease, are modelled using differential equations in science and engineering. Ordinary differential equations (ODEs) and partial differential equations (PDEs) are the two primary categories of differential equations.

To solve this differential equation using separation of variables, we first need to separate the variables Y and X on opposite sides of the equation:

dY / (Y + 1) = (2X + 1) dX

Following that, we incorporate both sides of the problem:

∫ dY / (Y + 1) = ∫ (2X + 1) dX

The integral on the left side can be evaluated using the substitution

u = Y + 1 and du = dY:

ln|Y + 1| = ∫ dY / (Y + 1) = ln |u| + C1

where C1 is the constant of integration.

The integral on the right side can be evaluated using the power rule of integration:

∫ (2X + 1) dX = X² + X + C2

where C2 is another constant of integration.

Putting these results together gives the general solution to the differential equation:

ln|Y + 1| = X² + X + C

where C = C1 + C2 is the combined constant of integration.

To solve for Y, we exponentiate both sides of the equation:

|Y + 1| = e⁽ˣ⁾2+X+C)

Taking into account the absolute value, we have two cases:

Case 1: Y + 1 = e⁽ˣ⁾2+X+C)

Y = e⁽ˣ⁾2+X+C) - 1

Case 2: Y + 1 = -e⁽ˣ⁾2+X+C)

Y = -e⁽ˣ⁾2+X+C) - 1

Therefore, the general solution to the differential equation DY/DX=(Y+1)(2X+1) is:

Y = e⁽ˣ⁾2+X+C) - 1 or Y = -e⁽ˣ⁾2+X+C) - 1

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Answer:

Step-by-step explanation:

To find the average mass of the six people, we need to divide the total mass by the number of people.

The total mass of the six people is 1/2 tonne, which is equivalent to 500 kg (since 1 tonne = 1000 kg).

So, the average mass of the six people is:

500 kg / 6 = 83.33 kg (rounded to two decimal places)

Therefore, the average mass of each person is approximately 83.33 kg.

The value of b when the equation is written in standard form is determined as: 6.

An equation can be expressed in standard form as:

Ax² + Bx + C = 0, where A, B, and C are constants.

Given that the situation is expressed by the equation, w² + 6w = 112, we can expressed this in standard form as follows:

w² + 6w = 112

Subtract 112 from both sides:

w² + 6w  - 112 = 112 - 112

w² + 6w - 112 = 0

The standard form is therefore, w² + 6w - 112 = 0, and the value of b is equal to 6.

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The expression 8x^3/2 × -7x^2/3 can be simplified to give

To reduce the given expression, we can employ the principles of exponents and carry out the multiplication.

The given expression is:- 8x^(3/2) * (-7x^(2/3))

Utilizing this property, combining two terms with exponents entails adding the exponents when they feature the same bases. Thus, we establish:

8 * (-7) = -56 (coefficient multiplication)

x^(3/2 + 2/3) results in x^(13/6) (exponent addition)

Bringing it all together, we have:

8x^(3/2) * (-7x^(2/3)) = -56x^(13/6)

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Part A: if you earned 14 points out of 17 points on the quiz, your percentage score would be 82.35%.

Part B: the grade earned for the quiz is a B.

What is the percentage?

A percentage is a means to represent a piece of 100 as a ratio or percentage. The sign for it is % (percent), which stands for "per hundred." If your state, for instance, that 50 out of 100 people like chocolate, then means that 50 out of 100 people, or 0.5 (50/100), like chocolate overall. In several disciplines, including finance, business, mathematics, and statistics, percentages are frequently used.

Here, we have

Given: Ms. Gray's social studies class.

A 90%–100%

B 80%–89%

C 70%–79%

Part A: Let's say we pick the number 17 as the number of points earned on the quiz. If the quiz is worth a total of 17 points, and you earned 14 points, then your percentage score is calculated as follows:

Percentage score = (points earned / total points) x 100%

Percentage score = (14 / 17) x 100%

Percentage score = 82.35%

Hence, if you earned 14 points out of 17 points on the quiz, your percentage score would be 82.35%.

Part B: Based on the percentage score of 82.35%, you would earn a B grade using the grading scale provided.

According to the grading scale, a B grade is earned for a percentage score between 80%-89%, and the percentage score obtained in part A is within this range.

Hence, the grade earned for the quiz is a B.

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Answer:

y = f(x) + 3

Hope this helps!

Step-by-step explanation:

Because the graph went up 3...

Answer:

y = f(x) + 3

Hope this helps!

Step-by-step explanation:

Because the graph went up 3...

Answer: -3 3/7

Step-by-step explanation:

First, you can convert -1 5/7 to an improper fraction by multiplying the denominator with the whole number, then add the answer to the numerator. Like this:

7 × 1 = 7

7 + 5 = 12

12 will be your numerator and 7 will be your denominator. When you're done converting, bring down the negative sign. The fraction should be:

-12/7

Now, since -1 5/7 is -12/7, we can divide.

To divide, we can use, KCF (Keep, Change, Flip)

First, keep -12/7.

Second, change division to multiplication.

Third, flip 1/2.

Your problem will be set up like:

-12 /7 × 2/1

Now, we can solve. Multiply both the numerator AND the denominator. Since you're multiplying, you don't have to change the denominator to match each other. The answer will be:

-24 / 7

However, this fraction is improper. We can use long division to make this fraction proper.

Divide 24 and 7. You'll get 3, with a remainder of 3.

The remainder will be the new numerator and the answer will be the whole number.

3 - Remainder (Whole Number)

3 - Answer (New Numerator)

7 - Dividend (Original Denominator) Will not be changed

Hence, your final answer is:

-3 3/7

Reply below if you have any questions or concerns.

You're Welcome!

- Nerdworm

[tex]-\dfrac{24}{7} }.[/tex]

[tex]\sf -1\dfrac{5}{7}=\\\\ \\ -1+\dfrac{5}{7}=\\\\ \\ -(\dfrac{7}{7}+\dfrac{5}{7})=\\ \\ \\-(\dfrac{12}{7})[/tex]

[tex]\dfrac{-\dfrac{12}{7} }{\dfrac{1}{2} }[/tex]

[tex]-\dfrac{12}{7} }*\dfrac{2}{1} =\\ \\ \\-\dfrac{12}{7} }*2=\\ \\ \\-\dfrac{12*2}{7} }\\ \\ \\-\dfrac{24}{7} }[/tex]

The town's emergency response planning committee has proposed to place four emergency response centers at the four corners of the town, with each center serving the people who live within 3 miles of the respective response center. The idea is to provide coverage to the entire town and ensure prompt emergency response for the residents. However, there are some problems with this idea.

Unequal coverage: Placing the emergency response centers at the four corners of the town may result in unequal coverage for the residents. Depending on the size, shape, and population distribution of the town, some areas may be farther away from the response centers, resulting in longer response times and reduced effectiveness in emergency situations.

Overlapping coverage: Placing four response centers in a small town may result in overlapping coverage areas, where the coverage areas of multiple response centers overlap with each other. This may lead to duplication of resources and inefficiencies in emergency response efforts.

Limited reach: Placing response centers only at the four corners of the town may result in limited reach for certain areas, especially those located in the middle or farther away from the corners. This may leave some residents outside the 3-mile coverage radius without access to timely emergency response services.

One potential solution to address these problems could be to use a more strategic approach to determine the locations of the emergency response centers. This could involve conducting a thorough analysis of the town's population density, geographical features, road network, and existing emergency resources. Based on this analysis, the response centers could be strategically placed at locations that provide the best coverage to the entire town, considering factors such as response time, resource allocation, and accessibility.

For example, instead of placing all the response centers at the corners of the town, they could be distributed more evenly across the town to ensure more equitable coverage. Additionally, the use of advanced GIS (Geographical Information System) technology and modeling techniques could help in identifying optimal locations for the response centers, taking into account various factors such as population density, road network, and travel time.

Furthermore, collaboration and coordination among the emergency response centers, along with proper communication and information sharing, can help in avoiding duplication of resources and improving the efficiency of emergency response efforts.

In conclusion, while the idea of placing four emergency response centers at the four corners of town may seem simple, there are potential problems such as unequal coverage, overlapping coverage, and limited reach. A more strategic and data-driven approach, considering factors such as population density, geographical features, and existing resources, can help in identifying optimal locations for the response centers and ensuring effective emergency response services for all residents of the town.

Answer:

  y = 2x

Step-by-step explanation:

You want the simplified form of y = |x -5| +|x +5| if x > 5.

The graph of the whole function will have turning points where the absolute value expressions are 0:

  x -5 = 0   ⇒   x = 5

  x +5 = 0   ⇒   x = -5

For values of x > 5, we are concerned with that portion of the graph that is to the right of both of these turning points. Hence, both absolute value expressions are positive and unchanged by the absolute value bars.

  y = (x -5) +(x +5) . . . . . . if x > 5

  y = 2x . . . . . . . . . . . . . . collect terms

The simplified function is y = 2x.

__

Additional comment

The attached graph shows y=2x and the given function for x > 5. They are identical. (The y=2x graph is shown dotted, so you can see the red graph of the given function.)

a) The rule that represents the height as a function of the number of bounces is described by H(n) = 1.5 · (71 / 100)ˣ⁻¹. (Correct choice: B)

b) The height at the top of the sixth path is equal to 0.27 meters. (Correct choice: B)

In this problem we need to derive the equation that represents maximum height as a function of the number of bounces:

H(n) = a · (r / 100)ˣ⁻¹

Where:

If we know that a = 1.5 m and r = 71, then the rule for the sequence is:

H(n) = 1.5 · (71 / 100)ˣ⁻¹

And the height at the top of the sixth path:

H(6) = 1.5 · (71 / 100)⁶⁻¹

H(6) = 0.27 m

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The expression is equal to 2 is 5 x (one-third x 6) ÷ 5. The correct option is C.

A mathematical expression is the collection of mathematical symbols that results from the proper combination of numbers and variables using operations like addition, subtraction, multiplication, division, exponentiation, and other as-yet-unlearned operations and functions.

Can be simplified each of the expressions see if them equals 2:

4 x (one-half x 6) ÷ 3 = 4 x 3 ÷ 3 = 4

6 ÷ (one-fourth x 3 x one and one-fourth) = 6 ÷ (3/4 x 5/4) = 6 ÷ (15/16) = 96/15 ≠ 2

5 x (one-third x 6) ÷ 5 = 5 x 2 ÷ 5 = 2

10 − (one-fifth x 10) + 1 = 10 - 2 + 1 = 9 ≠ 2

Therefore, the correct option is c. 5 x (one-third x 6)

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ANSWER:

They have $1,100 altogether.

EXPLANATION:

- Originally, Mike had $500, while Joel had $600. That adds up to $1,100, and it makes Joel have $100 more than Mike.

- Half of Joel’s $600 is $300, which he gave to Mike. That makes Joel now have $300 himself.

- Adding $300 to Mike’s $500 is $800, which means Mike now has $800.

- $800 (Mike’s new amount) minus $300 (Joel’s new amount) is $500, which works because Mike now has $500 more than Joel.

-$300 + $800 is, of course, still $1,100.

The total number of each zone for a three-week period is between 1200 and 1300, 1:2 is the ratio of customer from Zone 1 to customers from Zone 3.

​

1) During 3 weeks bound

Customers from zone 3 = 900

Customers from zone 4 = 320

So, total customers = (900 + 320)

= 1220

Hence, option D is correct i.e between 1200 and 1300.

2) Customer from zone 1 = 450

Customers from zone 3 = 900

So, Customer from zone 1 / Customer from zone 3 = 450/900

= 1/2

Hence, option 2 i.e., 1:2 is correct.

3) 3/5 pound of clay is used to make = 1 bowl

So, 1 bound of clay is used to make = 5/3 bowl

By unitary method,

10 pounds of clay is used to make = 10 *(5/3)

= 16.66

= 17 bowls (approx)

Hence, option d is correct answer.

Therefore, The total number of each zone for a three-week period is between 1200 and 1300, 1:2 is the ratio of customer from Zone 1 to customers from Zone 3.

​

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(a) 20% of $224,000 is $44,800.

(b) 3 points on a $224,000 mortgage is $6,720.

(c) $44,800 + $6,720 is $51,520. So the total down payment and closing costs are $51,520.

(d) The mortgage amount is $224,000 - $51,520 = $172,480.

(e)

Monthly payment = $172,480 * (7%/12%)(30 years) = $1,234.15

Total interest paid over 30 years = $172,480 * 7% * 30 years = $243,720

For the function in the graph the range is option B: 40≤f≤100.

The collection of all potential output values (y-values) that a function could produce is known as the function's range. It is, in other words, the entire set of values that the function is capable of accepting as its input changes across its domain. The collection of all potential output values (y-values) that a function could produce is known as the function's range. It is, in other words, the entire set of values that the function is capable of accepting as its input changes across its domain.

The range of the function is the output values, or the y-coordinates of the function.

For the given graph, the output of the function is from 40 to 100 thus, the range is:

40≤f≤100

Hence, for the function in the graph the range is option B: 40≤f≤100.

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If Thabang and Ndivhu decided to share their money according to hours, then Thabang will receive R240.

Thabang worked for 3 hours and Ndivhu worked for 2 hours.

Together, they worked for a total of 5 hours. They were paid R400 for their work,

So, we calculate their hourly-rate by dividing the total amount by the total number of hours worked;

⇒ Hourly rate = (Total amount paid)/(Total number of hours worked),

⇒ Hourly rate = R400/5,

⇒ Hourly rate = R80 per hour,

Now, we find amount Thabang will receive if it is decided to share the money according to "hours-worked".

Thabang worked for 3 out of the total 5 hours, so he should receive 3/5 of the total amount paid,

⇒ Amount Thabang will receive = (Hours Thabang worked / Total hours worked) × Total amount paid,

⇒ Amount Thabang will receive = (3/5) × R400,

⇒ Amount Thabang will receive = R240,

Therefore, Thabang will receive R240 if they decided to share their money according to hours worked.

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The given question is incomplete, the complete question is

Thabang and his friend Ndivhu share a job of cleaning a neighbors yard and they were paid R400, Thabang worked 3 hours and Ndivhu worked for 2 hours. Determine the amount Thabang will receive if they decided to share their money according to hours​.

The distance from the ball to the center of the green (CD) is approximately 13.2 yards.

Given:

AB = 110 yards (distance from marker to center of the green)

BC = 45 yards (distance from marker to golfer's ball)

angle BAC = 115° (angle formed when golfer turns towards the ball)

To find CD (distance from ball to center of the green):

Apply the cosine function to angle BAC:

cos(115°) = AB/BC  (∵cos θ = Adjacent side/ Hypotenuse)

Substitute the given values:

cos(115°) = 110/45

Calculate cos(115°):

cos(115°) ≈ -0.2924 (rounded to four decimal places)

Multiply both sides by BC to solve for AB:

AB = BC * cos(115°)

AB ≈ 45 * -0.2924

AB ≈ -13.16 yards (rounded to two decimal places)

Note: (The AB is negative, as the ball is moving in the opposite direction of the golfer's turn.)

CD = |AB| ≈ 13.16 yards (rounded to two decimal places)

CD ≈ 13.2 yards (rounded to one decimal point)

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Answer:

1.  Not true, so (5,2) is not a solution

2. Not true, so (-3,-6) is not a solution.

3. Not true, so (0, [tex]\frac{2}{3}[/tex]) is not a solution.

Step-by-step explanation:

Substitute 5 for x and 2 for y

1 - y > x - 6

1 - 2 > 5 - 6

-1 > -1

This is not true. -1 is not greater than -1

y < x - 1

Substitute -3 for x and -6 for y

2 - y [tex]\leq[/tex] 2x

2 - (-6) [tex]\leq[/tex] 2(-3)

8 [tex]\leq[/tex] -6  

This is not true. 8 is not less than -6.

y [tex]\geq[/tex] x

Substitute 0 for x and [tex]\frac{2}{3}[/tex]

3 - [tex]\frac{1}{2}[/tex] x + 3y < 8

3 - [tex]\frac{1}{2}[/tex] (0) + 3([tex]\frac{2}{3}[/tex]) < 8

3 - 0 + [tex]\frac{6}{3}[/tex] < 8

3 + 2 < 8

6 < 8

This is true. 6 is less than  8

y [tex]\geq[/tex] 1

[tex]\frac{2}{3}[/tex] [tex]\geq[/tex] 1

This is not true. [tex]\frac{2}{3}[/tex] is not greater than or equal to one.  

The ordered pair has to be a solution for both equations for the ordered pair to be a solution for the system.

Helping in the name of Jesus.

Answer:

Yes, because while the total number of professors increases, the number of pure math professors decreases.

Step-by-step explanation:

Using Hamilton's method:

pure math 4/28 = 0.142857

applied math 12/28 = 0.42857

statistics = 12/28 = 0.42857

10 × 0.142857 = 1.42857

10 × 0.42857 = 4.2857

10 × 0.42857 = 4.2857

numbers of professors:

pure math 1

applied math 4

statistics 4

total 9

Add 1 to pure math

Final original numbers of professors using Hamilton's method

pure math 2

applied math 4

statistics 4

total 10

Add 1 professor to department.

11 × 0.142857 = 1.5714

11 × 0.42857 = 4.71428

11 × 0.42857 = 4.71428

numbers of professors:

pure math 1

applied math 4

statistics 4

total 9

Add 1 to applied math and 1 to statistics

Final new numbers of professors using Hamilton's method

pure math 1

applied math 5

statistics 5

total 11

Despite the addition of 1 professor to the department, the field of pure mathematics went from 2 professors to 1 professor. This is an example of the Alabama paradox.

Answer: Yes, because while the total number of professors increases, the number of pure math professors decreases.

Step-by-step explanation:

We can use the two data points given to find the equation of the line, which gives the monthly cost (y) in terms of the calling time in minutes (x).

First, we can find the slope of the line:

slope = (change in y) / (change in x)

slope = (20.66 - 18.45) / (56 - 39)

slope = 0.219

Next, we can use one of the data points and the slope to find the y-intercept (b). Let's use the data point (39, 18.45):

y - y1 = m(x - x1)

y - 18.45 = 0.219(x - 39)

y - 18.45 = 0.219x - 8.541

y = 0.219x + 9.909

So the equation for the monthly cost is y = 0.219x + 9.909.

To find the monthly cost for 50 minutes of calls, we plug in x = 50:

y = 0.219(50) + 9.909

y ≈ $21.44

Therefore, the monthly cost for 50 minutes of calls is approximately $21.44.

(b) The transformation stretches the graph by a factor of 5 describes the effect of this transformation on the graph of the quadratic function along the y-axis.

The transformation from y = 2x² to y = 10x² changes the coefficient of the x² term from 2 to 10. This change in the coefficient affects the vertical scaling of the graph of the quadratic function along the y-axis. When we multiply the entire function by a constant, it causes a vertical stretch or compression of the graph depending on the magnitude of the constant.

In this case, the transformation stretches the graph along the y-axis by a factor of 5 since 10 is 5 times greater than 2. This means that the vertical distance between the points on the graph of the function is now 5 times greater than before the transformation.

Therefore, the correct option is (b) - The transformation stretches the graph by a factor of 5.

Correct Question :

If you transform y = 2x2 into y = 10x2, which option below describes the effect of this transformation on the graph of the quadratic function along the y-axis?

a) The transformation shrinks the graph by a factor of 25.

b) The transformation stretches the graph by a factor of 5.

c) The transformation stretches the graph by a factor of 25.

d) The transformation shrinks the graph by a factor of 5.

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Oliver withdraws an amount of $285.10 from his saving account and the interest is $26.60

Compound interest is a type of interest that is paid on the closing balance at the end of the previous year, which includes the interest paid in previous years.

His saving account earns 1.8% annually

The interest Oliver could have earned in five months:

Monthly interest = Annual interest ÷ 12

Monthly interest = 1.8% ÷ 12

Monthly interest = 0.018 ÷ 12 = 3/2000

A = P(1 + r)n

After five months = Principle × (1 + interest)ⁿ

After five months = 285.10 × (1 + 0.018)⁵

After five months = 311.70

Interest earned = 311.70 - 285.10 = $26.60

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Complete paragraph proof would be detailed proof.

Given that M is the midpoint of PK and PK ⊥ MB, we need to prove that △PKB is isosceles.

Proof,

Since M is the midpoint of PK, PM ≅ KM.

Also, since PK ⊥ MB, we have ∠PMB and ∠KMB are right angles.

Since right angles are congruent, we have ∠PMB ≅ ∠KMB.

Now, by the SAS congruence theorem, we have △PMB ≅ △KMB because they share side MB, and PM ≅ KM and ∠PMB ≅ ∠KMB.

Thus, we have BP ≅ BK because corresponding parts of congruent triangles are congruent.

Therefore, △PKB has two congruent sides and isosceles by definition.

Hence, we have proven that △PKB is isosceles.

Correct Question :

Complete the paragraph proof. Given: M is the midpoint of PK PK ⊥ MB Prove: △PKB is isosceles It is given that M is the midpoint of PK and PK ⊥ MB. Midpoints divide a segment into two congruent segments, so PM ≅ KM. Since PK ⊥ MB and perpendicular lines intersect at right angles, ∠PMB and ∠KMB are right angles. Right angles are congruent, so ∠PMB ≅ ∠KMB. The triangles share MB, and the reflexive property justifies that MB ≅ MB. Therefore, △PMB ≅ △KMB by the SAS congruence theorem. Thus, BP ≅ BK because . Finally, △PKB is isosceles because it has two congruent sides.

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