All Answers 358516 Answers

-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

You made a mistake in this step:

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

(1/4)c should be: (1/2)c

The correct answer should be: c = 6 gallons.

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)

1. This means that the triangle is right and AB and BC are perpendicular, so use the formula 1/2 * b * h to get 1/2 * 6 * 8 = 3 * 8 = 24

2. The three angles of a triangle add up to 180, if we add 135 and 30 we get 165, and if we subtract that from 180 we get an answer of 15 degrees.

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

If we set up an equation where x is the amount of gallons that the tank can hold, then 1/6x + 2 = 1/2x. Try solving from there, if you need anything, just ask :)

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

5/9

Step-by-step explanation:

If you draw the triangle out on a grid papar and draw a line across the 2 marker point from XY up of the YZ side, you will see that the line from that marker point across to the hypotenuse is actually 8. Then you find the area of the little triangle with legs 8 and 4 (since 6-2 is 4), and the area is 16. After that, find the total area of the big triangle, which is 12x6/2=36. Since we want to know the area of the quadrilateral at the bottom of the big triangle seperated by the 2 marker point, we subtract 16 from 36, which equals to 20. 20/36=5/9. And that is my final answer.

The reason why it is so good is because this calculator provides everything from:

- Calcluator 

- And questions

Making it one of the best calculators and coolest people you will see here which is just amazing!

The smallest integer that Pearl wrote down is 15.

Step-by-step explanation:

7 consecutive integers. They are:

a

a+1

a+2

a+3

a+4

a+5

a+6

Sum of the integers:

a + a + 1 + a + 2 + a + 3 + a + 4 + a + 5 + a + 6 = 7a + 21

6 times the largest of the seven integers.

The largest is a + 6. So

7a + 21 = 6(a + 6)

7a + 21 = 6a + 36

7a - 6a = 36 - 21

a = 15

The smallest integer that Pearl wrote down is 15.

We know that the angle bisector theorem and the specified requirements are satisfied if BC = 16, CZ = 8, and BZ = 24. The angle bisector theorem is also satisfied if BC = 8, which also makes CZ = 16/3 and BZ = 112/9.

What is angle bisector theorem?

The opposite side of a triangle is divided into two portions that are proportional to the sides they are on if the angle is bisected by a line, according to the angle bisector theorem.

The angle bisector theorem, which asserts that if a line bisects a triangle's angle, it divides the opposite side into two segments that are next sides to each other, can be used to solve this problem.

Let BC and BZ both equal x. The angle bisector theorem provides us with:

AC/AB = CZ/BZ

16/(16+x) = 16/y

We solve y in terms of x, and as a result,

y = 16(16+x)/16 = x+16

Upon substitution, we discover—

16/(16+x) = 16/(x+16)

If we cross-multiply, we obtain:

16(x+16) = 16(16+x)

Simplifying, we get:

16x + 256 = 256 + 16x

In light of the fact that 16x cancels on both sides, we are left with:

256 = 256

As a result, there exist an endless number of BC and BZ values that could satisfy the requirements.

28

Step-by-step explanation 

In order to solve this problem we need to find out what $QR$ is.

We know that the perimeter of the parallelogram was given as $24$ and that two of the sides were given as $5$. $5+5=10$ $24-10=14/2=7$. Now we know that the base was $7$.

We can find the height by taking the area that they gave us and dividing that by $7$. We get $4$.

Since we now know that the height of the parallelogram is $4$ that means $OQ$ is also $4$.

Next we need to find what $RT$ is.  

$4^2+B^2=5^2$.

Since we now $5^2$ is $25$ and that $4^2$ is $16$, we can use the following equation.

$25-16=9$. $\sqrt{9}=3$.  

Now we know that $RT$ is $3$ we can start adding the sides.

$7+3=10+4=14 \cdot 2=28$.

[b]The perimeter of the rectangle $QOST$ is 28![/b]

71.4% of the students bring an umbrella only.

What is the percentage?

Percentage is a way of expressing a number as a fraction of 100. It is represented by the symbol %. It is a useful way of comparing values and calculating changes over time. 

Let's denote the total number of students in Margo's school by the letter N. We can use a Venn diagram to represent the number of students who bring a raincoat, an umbrella, or both, as shown below:

           RAINCOAT         UMBRELLA

          ________         ________

         |                  |       |                 |

         |                  |       |                 |

         |                  |       |                 |

         |      A         |       |        C       |

         |                  |       |                 |

         |                  |       |                 |

         |________|       |________|

          BOTH (B)         ONLY (D)

We know from the problem statement that 50% of the students bring both a raincoat and an umbrella. Therefore, the number of students who bring both is equal to 0.5N.

We also know that the total number of umbrellas brought is twice the number of raincoats. Let's denote the number of raincoats brought by the letter A. Then the number of umbrellas brought is equal to 2A.

We can use these values to fill in the Venn diagram. The total number of students is N, so the number of students who bring only a raincoat is (A - 0.5N) and the number of students who bring only an umbrella is (2A - 0.5N). The number of students who bring both is 0.5N.

To find the percentage of students who bring an umbrella only, we need to divide the number of students who bring only an umbrella by the total number of students and multiply by 100:

(2A - 0.5N) / N * 100

We can simplify this expression using the fact that the number of umbrellas brought is twice the number of raincoats:

(2A - 0.5N) / N * 100

= (2 * A - 0.5N) / N * 100

= (3A - 0.5N) / N * 100

We can't directly solve for the value of A, but we can use the fact that the sum of the numbers in the Venn diagram adds up to N:

A - 0.5N + 0.5N + 2A - 0.5N = N

3.5A - N = N

A = 2N / 7

Substituting this value of A into the expression for the percentage of students who bring an umbrella only, we get:

(3A - 0.5N) / N * 100

= (3 * 2N / 7 - 0.5N) / N * 100

= 71.4%

Therefore, 71.4% of the students bring an umbrella only.

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.

By my guess you wanted written as a fraction. :

Written as a fraction:

\({x-2 \over 3=5}\)

The first appearance of the integral symbol in print was in a paper by Leibniz in the Acta Eruditorum. The integral symbol was actually a long letter S for "summa." In his Quadratura curvarum of 1704, Newton wrote a small vertical bar above x to indicate the integral of x.

Vincent, an artist, plans to paint and sell some miniature paintings. He just bought some brushes for $50, and paint and canvas for each painting costs $5; he will sell each painting for $10. Once Vincent sells a certain number of his paintings, he will be breaking even. How many paintings will that be?

Value of D is 55 us dollars

Step-by-step explanation:

So first you would figure out 1010-350 to find out the cost just for those 12 months without to set up fee.

Then you would divide 12 by y 660 to see how much is the additional cost per month.

Therefore, the side length of the cube is 1/3, as required.

What is tetrahedron?

Four triangular faces make up the three-dimensional shape of a tetrahedron. The foundation of the pyramid is one of the triangles, with the other three triangles joining it.

Let D be the vertex of the cube on face ABC. 

Since the opposite vertex of the cube is at O, we have OD = 1. 

Let the side length of the cube be x.

Consider triangle AOB.

AB² = AO² + OB² = 1 + 1 = 2

Similarly, find that BC² = AC² = 2.

Since ABC is a right triangle with angles A, B, and C being 90° -

sin A = BC / AB = √2 / 2

sin B = AC / AB = √2 / 2

sin C = BC / AC = 1

Consider tetrahedron ABCO. Since AOB, AOC, and BOC are right angles -

∠AOCB = π - ∠AOC - ∠BOC = π/2

∠AOBC = π - ∠AOB - ∠BOC = π/2

∠ABCO = π - ∠AOC - ∠AOB = π/2

So triangles AOC, AOB, and BOC are all right triangles with hypotenuse 1 and angles A, B, and C, respectively. 

Using the sine rule -

sin AOC = AO / OC = 1

sin AOB = sin BOC = BO / OC = 1

Therefore, the areas of triangles AOC, AOB, and BOC are -

Area(AOC) = (1/2) × AO × OC × sin AOC = (1/2) × 1 × 1 × 1 = 1/2

Area(AOB) = Area(BOC) = (1/2) × BO × OC × sin AOB = (1/2) × 1 × 1 × 1 = 1/2

Now, consider triangle AOD.

sin AOD = sin(180° - AOB - AOC) = sin(BOC) = √2 / 2

Using the sine rule -

AD / sin AOD = OD / sin OAD

AD / (√2 / 2) = 1 / x

AD = (√2 / 2) * (1 / x)

The area of triangle AOD is -

Area(AOD) = (1/2) × AD × OD × sin AOD = (1/2) × (√2 / 2) × (1 / x) × 1 × (√2 / 2) = 1 / (2x²)

Now, consider the tetrahedron ABCO. 

The volume of the tetrahedron is -

V = (1/3) × Area(ABC) × OD = (1/3) × (√3 / 4) × 1 = √3 / 12

The volume of the cube is -

V = x³

Since the cube is inscribed in the tetrahedron -

√3 / 12 = x³

So, now there is -

x = 1/3

Therefore, the side length of the cube is 1/3, as required