What is the center and radius of the circle with equation x2 + y2 - 4x + 22y + 61 = 0?

Final answer:

The quotient when dividing 6x³ + 13x² + x - 2 by x - 1 is 6x² + 19x + 20, and the remainder is 18. This solution can be found using synthetic division.

Explanation:

The question asks to find the quotient and remainder when the polynomial 6x³ + 13x² + x - 2 is divided by x - 1. To solve this, we can use polynomial long division or synthetic division. As the divisor is in the form of x - a, where a is 1, synthetic division is a streamlined approach.

We set up our synthetic division with 1 as our key number outside the division symbol and the coefficients of our polynomial (6, 13, 1, -2) inside:

Bring down the 6.

Multiply 1 by 6 to get 6, add this to the next coefficient 13 to get 19.

Continue the process: Multiply 1 by 19 to get 19, add this to 1 to get 20.

Repeat: Multiply 1 by 20 to get 20, add this to -2 to get 18.

The numbers now inside the division symbol represent the coefficients of the quotient polynomial, giving us 6x² + 19x + 20 with a remainder of 18. Therefore, the quotient is 6x² + 19x + 20, and the remainder is 18.

Answer:  2

Step-by-step explanation:

Let Set A represents the number of employees cooked​ food.

Set B represents the number of employees washed​ dishes.

Set C represents the number of employees operated the cash​ register .

Given : n(A)=9 ; n(B)=14 ; n(C)=22

n(A∩B)=4 ; n(A∩C) = 2 ; n(B∩C)=7 ; n(A∩B∩C)=1

Using the given information, we have made a Venn diagram below.

Using Venn diagram , we have

n(only A)= n(A)-(n(A∩B)+n(A∩C)+n(A∩B∩C))

⇒n(only A)=9-(4+2+1)=9-7=2

Hence, 2 employees only cooked​ food.

The length of the hypotenuse is 20 centimeters.

In order to solve this question, we'll use the Pythagoras formula which will be:

a² + b² = c²

where,

First leg = a = 12 centimeters

Second leg = b = 16 centimeters

Hypothenuse = c = Unknown

It should be noted that the hypothenuse is the longest side and therefore the formula will be:

a² + b² = c²

12² + 16² = c²

144 + 256 = c²

400 = c²

Therefore, c = ✓400

c = 20

Therefore, the length of the hypotenuse is 20 centimeters.

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The two positive numbers are 10 and 5.

Given that,

The difference between two positive numbers is 5.

And the sum of their squares is 125.

Let us assume that,

The numbers are x and y.

Here, the difference between two positive numbers is 5.

x - y = 5

x = 5 + y  .. (i)

And, the sum of their squares is 125.

That is,

x² + y² = 125

Substitute the value of x from (i);

(5 + y)² + y² = 125

25 + y² + 10y + y² = 125

2y² + 10y + 25 = 125

2y² + 10y - 100 = 0

2 (y² + 5y - 50) = 0

y² + 5y - 50 = 0

y² + (10 - 5)y - 50 = 0

y² + 10y - 5y - 50 = 0

y (y + 10) - 5 (y + 10) = 0

(y - 5) (y + 10) = 0

This gives,

y = 5

y = - 10

Since the numbers are positive.

Hence, y = - 10 is not possible.

From (i);

x = 5 + y

Put y = 5;

x = 5 + 5

x = 10

Hence,

The two positive numbers are 10 and 5.

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The complete data set will be 2, 5, 7, 9, 9, 9, 12. When the number 9 is added to the collection, the median and mode are both 9.

The median is defined as the value range. To get it, arrange the numbers in ascending order from smallest to largest, then hide one number on either end until you reach the center.

The median of a set is the middle number when the numbers are arranged in numerical order.

The mode of a set is the number that appears most frequently.

To find an entry that will make the median and mode equal, we first need to arrange the given numbers in numerical order: 2, 5, 7, 9, 9, 12.

The median of this set is the middle number, which is 9.

The mode is 9, since it appears twice while the other numbers appear only once.

To make the median and mode equal, we need to find a number that is between 1 and 12 and that is equal to 9. The only such number is 9.

Therefore, if we add the number 9 to the set, the median and mode will both be 9. The complete data set will be 2, 5, 7, 9, 9, 9, 12.

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Answer:

B. The graph of g is the graph of f translated to the left 3 units and down 7 units.

Step-by-step explanation:

f(x) = x²  to g(x) = (x + 3)² - 7

Notice the difference:

from 'x' to 'x + 3'

and then from 'x²' to '(x+3)² - 7'

The transformation that changes 'x' to 'x + 3' is the translation across the x-axis, three units to the left. The transformation that changes 'x²' to (x + 3)² - 7' is the translation across the y-axis, seven unit down. Refer to the graph attached below to visualize the transformation. Take the vertex of f(x) = x² and translate it three unit left and seven unit down to get g(x).

PART A:

The function in vertex form by completing the square is:

            [tex]f(t)=(t+3)^2-29[/tex]

PART B:

The vertex is located at (-3,-29)

and the vertex is the minimum point of the graph.

PART C:

      The axis of symmetry of f(t) is:

                    t= -3

The function f(t) is given by:

             [tex]f(t)=t^2+6t-20[/tex]

We know that the vertex form of a equation is given by:

           [tex]f(t)=a(t-h)^2+k[/tex]

where a,h and k are real numbers.

and the vertex is located at (h,k)

The function is given by:

        [tex]f(t)=(t+3)^2-29[/tex]

Hence, the vertex is given by:

            (-3,-29)

Also, as the leading coefficient is positive.

Hence, the parabola is upward open parabola.

Hence, the vertex will be the minimum point of the graph.

Also, the axis of symmetry of a graph is given by:

                          t=h

Hence, here the axis of symmetry is:  t= -3

Answer:

[tex]30-(1.79a+0.36b+2.99)[/tex]

Step-by-step explanation:

Let  a represents the number of pounds of apples Abigail buys

Cost of 1 pound of apple = $1.79

So, Cost of a pounds of apple = 1.79a

Let b represents the number of bananas she buys,

Cost of 1 pound of banana = $0.36

Cost of b pounds of banana = 0.36b

Abigail buys 3 pounds of potatoes

Cost of 3 pounds of potatoes = $2.99

So, Total cost = [tex]1.79a+0.36b+2.99[/tex]

Abigail has $30.00 to buy groceries.

So, the amount of change she receives after her purchases=[tex]30-(1.79a+0.36b+2.99)[/tex]

So,Option C is correct.

Hence The  expression represents the amount of change she receives after her purchases is [tex]30-(1.79a+0.36b+2.99)[/tex]

the equation for a line is y = mx+b

 m = slope

b = y intercept

 you know the slope is 2/3 and the y intercept is 7 so just replace them in the equation:

 y =2/3x +7

To calculate the maximum speed a subway train can reach between stations, given the maximum tolerable acceleration, we use the equation Vmax = sqrt(2 × a × d), resulting in a maximum velocity of approximately 40.21 m/s.

To find the maximum speed a subway train can attain between stations, given a maximum tolerable acceleration for passengers, we need to use the equations for uniformly accelerated motion. Since the train starts and ends at rest at the stations and because the stations are 930 meters apart, we can think of the train's trip as two identical parts: an acceleration phase and a deceleration phase.

Let's denote the maximum speed as Vmax, the acceleration as a, and the distance for each phase as d (half the distance between stations). We have:

Using the equation Vmax2 = 2ad, we can solve for the maximum speed (Vmax) as follows:

Vmax = sqrt(2 × a × d)

Substituting the values, we get:

Vmax = sqrt(2 × 1.74 m/s2 × 465 m)

Vmax ≈ sqrt(1617) m2/s2 ≈ 40.21 m/s

Therefore, the maximum speed the subway train can attain between stations is approximately 40.21 m/s.

The  solution of the given equation is 5.5.

A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions.

The given equation is 13=4x-9.

The collection of values used to replace the unknowns in an equation to make it true is known as the solution.

Here, add 9 on both the sides of an equation.

That is, 13+9=4x-9+9

4x=22

Divide 4 on both the sides of an equation.

4x/4 =22/4

x=5.5

Therefore, the value of x is 5.5.

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11(y-10) = 4y-5

11y-110 = 4y-5

7y -110 = -5

7y =105

y = 105/7 = 15

PTS = 11(15-10) = 11*5 = 55 degrees

Answer:

55

Step-by-step explanation:

We are given with two lines that intersects each other .

the vertical opposite angles are same.

So we equate the angles and solve for y

[tex]11(y-10)= 4y-5[/tex]

[tex]11y-110=4y-5[/tex]

Subtract 4y from both sides

[tex]7y-110=-5[/tex]

Now we add 110 on both sides

[tex]7y=105[/tex]

Now we divide by 7 on both sides

y=15

now we find m∠PTS

m∠PTS=[tex]11(y-10)[/tex]

Plug in 15 for y

m∠PTS=[tex]11(15-10)=55[/tex]

The maximum possible length it could have been 44.49cm.

Rounding off means a number is made simpler by keeping its value intact but closer to the next number. It is done for whole numbers, and for decimals at various places of hundreds, tens, tenths, etc.

here, we have,

44cm is rounded to the nearest whole cm.

as, it is rounded to the nearest whole cm.

so, we get,

The maximum possible length it could have been 44.49cm.

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