All Questions ⁺⁰231021 Questions

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Hexagon

A regular hexagon has a perimeter of $p$ (in length units) and an area of $A$ (in square units). If $A = 3,$ then find the side length of the hexagon.

Akhain1 Mar 29, 2024

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Polygons

Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.
An equilateral triangle, a regular dodecagon, and a regular -gon, all with the same side length, also completely read more ..

Akhain1 Mar 29, 2024

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Diagonals

The number of diagonals in a certain regular polygon is equal to $2$ times the number of sides. How many sides does this polygon have?

Akhain1 Mar 29, 2024

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geometry

A sphere of radius $4$ inches is inscribed in a cone with a base of radius $8$ inches. In inches, what is the height of the cone? Express your answer as a decimal to the nearest tenth.

Akhain1 Mar 29, 2024

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geometry

Rose has a spherical plum of radius $2$ and a spherical watermelon of radius $8$. She builds a glass sphere around the two fruits to contain them, making the sphere as small as possible. When she has done this, she places a golf ball inside read more ..

Akhain1 Mar 29, 2024

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help geometry

A sphere is inscribed in a cone with height $3$ and base radius $3$. What is the ratio of the volume of the sphere to the volume of the cone?

Akhain1 Mar 29, 2024

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+152

HELP

Consider the quadratic equation ax^2 + bx + c = 0 with a, b, c integers from -4 to 4 and a is not 0. The solutions to this equation are in the form where D is square free . Determine the largest possible value of D.

Lilliam0216 Mar 29, 2024

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Constants

Find constants A and B such that
(2x + 11)/(x^2 - 6x + 5) = A/(x - 3) + B/(x + 4)
for all x such that x neq 7 and x neq 8

LiIIiam0216 Mar 29, 2024

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+956

Help with floor

Let f(x) = lfloor (2 - 3x)/(3x + 8) \rfloor
Evaluate $f(1)+f(2) + f(3) + \dots + f(999)+f(1000).$ (This sum has $1000$ terms, one for the result when we input each integer from $1$ to $1000$ into $f$.)

LiIIiam0216 Mar 29, 2024

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Hard geometry probs

Don't be dicouraged if you can't solve these
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2.
<#> read more ..

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Akhaim1 Mar 29, 2024

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Inequalities

(1) Let a_1, a_2, a_3 be real numbers such that
|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_1| = 1.
What is the largest possible value of |a_1 - a_2|?

(2) Let a_1, a_2, a_3, \dots, a_{10} be real numbers such that read more ..

LiIIiam0216 Mar 29, 2024

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+956

Absolute value

Find all solutions of the equation
|x^2 - 30x - 1| + |x^2 - 30x + 29| = 30

LiIIiam0216 Mar 29, 2024

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Please Help! Thank you

The largest term in the binomial expansion of is of the form , where and are relatively prime positive integers. What is the value of ?
Please help! Thank you so much! All I know is to use the binomial read more ..

greenplanet2050 Mar 29, 2024

Mar 28, 2024

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Polygon

Let $IJKLMN$ be a hexagon with side lengths $IJ = LM = 3,$ $JK = MN = 3,$ and $KL = NI = 3$. Also, all the interior angles of the hexagon are equal. Find the area of hexagon $IJKLMN$.

Akhain1 Mar 28, 2024

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Quadrilateral

In the diagram below, each side of convex quadrilateral $ABCD$ is trisected. (For example, $AP = PQ = QB.$) The area of convex quadrilateral $ABCD$ is $180.$ Find the area of the shaded region. read more ..

Akhain1 Mar 28, 2024

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Polygons

Let $B,$ $A,$ and $D$ be three consecutive vertices of a regular $20$-gon. A regular heptagon is constructed on $\overline{AB},$ with a vertex $C$ next to $A.$ Find $\angle BAD,$ in degrees.

Akhain1 Mar 28, 2024

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Quadrilateral

Quadrilateral $ABCD$ is a parallelogram. Let $E$ be a point on $\overline{AB},$ and let $F$ be the intersection of lines $DE$ and $BC.$ The area of triangle $EAD$ is $9.$ If $AE:EB = 3:4$ and the height of parallelogram $ABCD$ is $10$, read more ..

Akhain1 Mar 28, 2024

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help geometry

In triangle $ABC$, $M$ is the midpoint of $\overline{BC}$, and $N$ is the midpoint of $\overline{AC}$. The perpendicular bisectors of $BC$ and $AC$ intersect at a point $O$ inside the triangle. If $\angle AOB = 90^\circ$, then find the measure read more ..

Akhain1 Mar 28, 2024

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+51

halp!

At a meeting, 4 mathematicians, 3 scientists, and 2 journalists are to be seated around a circular table. How many different arrangements are possible if the mathematicians must all sit together (in 4 consecutive seats) and the 3 scientists must read more ..

fjeihqo0 Mar 28, 2024

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help plz

In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven numbers (1, 2, 3, 4, 5, 6, and 7). Among all the cards of each color, there is exactly one card labeled with each number. read more ..

fjeihqo0 Mar 28, 2024

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help

Four adults and four students are to be seated at a circular table. In how many different ways can they be seated if each adult must be next to two students? (Two seatings are considered the same if one can be rotated to form the other.) read more ..

fjeihqo0 Mar 28, 2024

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How are linear velocity and angular velocity related?

How are linear velocity and angular velocity related?
Drag a word, phrase, or equation into each box to correctly complete the statements.
Angular velocity is the change in distance over time, where distance is _____.
Angular read more ..

xXValeXx Mar 28, 2024

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Angular and Linear Velocities

Two wheel gears are connected by a chain. The larger gear has a radius of 8 centimeters and the smaller gear has a radius of 3 centimeters. The smaller gear completes 24 revolutions in 20 seconds.
What is the linear velocity of each read more ..

xXValeXx Mar 28, 2024

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Geometry

Points $A$, $B$, and $C$ are on a circle such that $AB = 8$, $BC = 15$, and $AC = 12$. Find the radius of the circle.

Akhain1 Mar 28, 2024

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Geometry

In triangle $ABC$, let $I$ be the incenter of triangle $ABC$. The line through $I$ parallel to $BC$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. If $AB = 5$, $AC = 5$, and $BC = 8$, then find the area of triangle $AMN$.

Akhain1 Mar 28, 2024

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