acyclics

-a^3 + 2a^2 - 4a + 3

( Not so ) step-by-step explantion

subtract like terms 

5a^3 - 6a^3 = -a^3

7 - 4 = 3

= -a^3 + 2a^2 - 4a + 3

112,500

Step-by-step explanation:
112,500 are all 6-digit multiples of 8 and NOT the sum of the digits that are divisible by 8. The number of all 6-digit integers where the sum of the digits is divisible 8 ==112,496

In the triangle PQR, let X represent the point where angle P and side QR connect, and let Y represent the foot of the perpendicular that runs from X to side PR. The length of XY is  2.25 × √(3) units.

What are triangles?

By drawing straight lines from three non-collinear points, a triangle is a three-sided polygon.

It is a basic geometric shape having several properties and applications in science, technology, engineering, and other fields.

Depending on the size of their angles and side lengths, triangles can be divided into different categories.

Due to the fact that PQ = QR = PR, triangle PQR is an equilateral triangle.

Let's write x as the length of each angle in this triangle. Since the angle P is divided into two equal halves by the angle bisector, the measures of the angles PXQ and QXR are both x/2.

Let's write d to represent XY's length. Trigonometry can be used to determine the length of XY since triangle PXY is a right triangle. Specifically, we have

tan(30) = XY / PY

Since PY = PQ - QY and PQ = QR = 9, we have PY = 9 - QY. Therefore:

tan(30) = XY / (9 - QY)

Solving for XY, we get:

XY = (9 - QY) tan(30)

QY needs to be located. The Pythagorean theorem can be used to determine QY because triangle QYX is also a right triangle:

QY² + XY² = QX²

Triangle QRX being a 30-60-90 triangle gives us:

QX = QR / 2 = 4.5

Therefore:

QY² + XY² = 4.5²

QY² + (9 - QY)² tan²(30) = 4.5²

The result of simplifying and solving for QY is:

QY = 2.25

Inputting this value into the XY expression yields the following result:

XY = (9 - 2.25) tan(30) = 2.25 × √(3)

Lets O be the center of the circle. From point O, draw a line perpendicular to XY such that it will intersect the circle at Z. That will be the greatest possible perimeter of triangle XYZ. 

Area of the circle = (pi)r^2 = 18pi

so, r = 3sqrt2 = OX = OY = OZ (all are radius of the circle)
YZ^2 = OY^2 + OZ^2 = (3sqrt)^2 + (3sqrt)^2 = 36
YZ = 6

diameter, XY= 6sqrt2
XZ = YZ = 6
so perimeter = 6 + 6 + 6sqrt2 = 12 + 6sqrt2. 

64%

Step-by-step exaplanation:

64% of the kids only bring an umbrella. Set up a system of equations: suppose x% of kids only bring raincoats, and y% only bring umbrellas. Since 8% of kids bring both, 1-8%=92% bring only one of the two, so x+y=92. Percent of kids who bring raincoats is x+8, and percent of kids bringing umbrellas is y+8. y+8=2(x+8). Solve the two equations, we get x=28, y=64. 64% kids only bring umbrellas.

(-1.6, 0.8)

Step-by-step explanation:

Given, equation of the line is y = 2x + 4 --- (1)

Closest point from origin will be the perpendicular distance from origin to the line.

We need to find out equation of the perpendicular from (0,0) on y = 2x + 4.

The equation is in slope intercept form i.e. y = mx + c

Slope, m = 2

Slope of the perpendicular = -(1/m) = -1/2

Equation of the perpendicular is found by (y - y1) = m (x - x1)

⇒ y - 0 = (-1/2) (x - 0)

⇒ y = (-1/2)x

⇒ 2y + x = 0 --- (2)

Solving (1) and (2), we get,

⇒ 5x = -8

⇒ x = -8/5

Substiute x = -8/5 in eq(2), we get

⇒ y = 4/5

x = -1.6 and y = 0.8

Therefore, the point on the line y = 2x + 4 closest to origin is (-1.6, 0.8)

The values of k that allow us to factor it as a product of linear binomials with integer coefficients are k ≥ -16√3 

What are linear binomials?

Linear binomials are mathematical expression that contain only two terms. Examples include x + 3.

What is a quadratic equation?

A quadratic equation is an equation in which the highest power of unknown is 2.

How to find the values of k?

Since we have the quadratic equation 23x² + kx - 5 + x² - 3, to determine the value of k which makes it a linear binomial, we simplify and find the discriminant.

What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation ax² + bx + c is D = b² - 4ac

Simplifying the quadratic equation, we have

23x² + kx - 5 + x² - 3 = 23x² + x² + kx - 5 - 3

= 24x² + kx - 8

Comparing the above quadratic equation with ax² + bx + c, we have that

a = 24

b = k and 

c = -8

So the discriminant, D =  b² - 4ac

= k² - 4 × 24 × 8

= k² - 768

The allowable values of k are gotten when the discriminant D ≥ 0

So, D ≥ 0

⇒ k² - 768 ≥ 0

⇒ k² ≥ 768

⇒ k ≥ ±√768

⇒ k ≥ ±√(256 × 3)

⇒ k ≥ ±16√3

⇒ k ≥ -16√3 

So, the values of k are k ≥ -16√3

* I have answered this before *

The tank capacity is 24 gallons

Step-by-step explanation:

Let the tank capacity be c.

Then (1/6)c + 2 = (1/4)c, or (since the LCD is 12):

2c/12 + 24/12 =  3c/12, or

24/12 = c/12

Multiplying both sides by 12, we get:

24 = c

The tank capacity is 24 gallons.

Here: https://web2.0calc.com/questions/fractions_75591

Three pairs of siblings from different families can be seated in two rows of three chairs so that siblings may not sit next to each other in the same row, and no child may sit directly in front of their sibling in 96 ways.

There are 3 pairs of siblings. We can name them as a₁, a₂, b₁, b₂, c₁ and c₂, a total of 6 children.

There are 2 rows of 3 seats each.

We will try to find the number of ways each seat can be filled.

So considering the first seat in first row, it can be filled in 6 different ways. Because any one from 6 children, say a₁, can be seated.

Now for the second seat in first row can only be filled by one of b₁, b₂, c₁ or c₂ because the first seated child a₁ along with the sibling a₂ of the same cannot be seated in the second seat. So in 4 different ways.

For the third seat remaining in the first row, one out of the remaining  pair of siblings c₁ or c₂ can be seated. This can be done in 2 ways.

Now after filling the first row in this manner, the second row can be filled in only 2 ways.

So total number of ways the children can be seated = 6 x 4 x 2 x 2 = 96 ways

angle C = 38 degrees

Step-by-step explanation:

Refer to attached figure (sorry, forgot to attach earlier)

Given 

AIB = 109

Let

CAX = XAB = x

CBY = YBA = y

XIB = YIA = x+y    ........exterior angles

XIB = YIA = 180-109 = 71   ............ sum of angles on a line

=>

x+y = 71

ACB = 180 - 2x -2y  ................. sum of angles of a triangle

= 180 - 2(x+y)

= 180 - 2(71) 

= 180 - 142

= 38