All Questions ⁺⁰235782 Questions

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Algebra

Evaluate a^3 - \dfrac{1}{a^3} if a - \dfrac{1}{a} = 0.

crimefightingvigiI Nov 19, 2024

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+245

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence. If
a_n = a_{n - 1} + a_{n - 2}
for all $n \ge 3,$ and $a_{11} = 1$ and $a_{10} = 4,$ then find $a_6.$

crimefightingvigiI Nov 19, 2024

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+843

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?

AnswerscorrectIy Nov 19, 2024

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+843

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 ..

AnswerscorrectIy Nov 19, 2024

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+843

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.

AnswerscorrectIy Nov 19, 2024

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+631

Geometry

Point $D$ is the midpoint of median $\overline{AM}$ of triangle $ABC$. Point $E$ is the midpoint of $\overline{AB}$, and point $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$. Find the area of triangle $BET$ if $AB = 20$, $AC = 20$,
<read more ..

siIviajendeukie Nov 19, 2024

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+631

Geometry

Let $M$, $N$, and $P$ be the midpoints of sides $\overline{TU}$, $\overline{US}$, and $\overline{ST}$ of triangle $STU$, respectively. Let $\overline{UZ}$ be an altitude of the triangle. If $\angle TSU = 60^\circ$ and $\angle STU = 45^\circ$, read more ..

siIviajendeukie Nov 19, 2024

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+631

Geometry

In triangle $ABC$, $\angle A = 30^\circ$ and $\angle B = 90^\circ$. Point $X$ is on side $\overline{AC}$ such that line segment $\overline{BX}$ bisects $\angle ABC$. If $BC = 12$, then find the area of triangle $BXA$.

siIviajendeukie Nov 19, 2024

Nov 18, 2024

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+24

Plz help asap!

Hi I would appreciate it if someone helps me solve the question: Factor as much as possible. Thanks!!

crimefightingvigil Nov 18, 2024

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+1557

Geometry

In triangle ABC, the angle bisector of angle BAC meets BC at D, such that AD = AB. Line segment AD is extended to E, such that angle DBE = angle BAD = 17 degrees. Find angle ABD.

parmen Nov 18, 2024

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+1557

Geometry

In triangle ABC, angle ACB = 90 degrees. Let H be the foot of the altitude from C to side AB.
If BC = 1 and AH = 2, find CH.

parmen Nov 18, 2024

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+1557

Geometry

Let ABC be a triangle, and let its angle bisectors be AD, BE, and CF which intersect at I. If DI=3, BD=4 and BI=6 then compute the area of triangle BID.

parmen Nov 18, 2024

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+842

Algebra

Let a and b be positive real numbers such that a + b = 1. Find the set of all possible values of a/b.

magenta Nov 18, 2024

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+842

Algebra

A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to five times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right read more ..

magenta Nov 18, 2024

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+842

Algebra

The graphs of x = y^4 and x + y^2 = 1 + x^2 - y^3 intersect at two points. The distance between these points is of the form sqrt(u + v*sqrt(5)), where u and v are integers. Find the ordered pair (u,v).

magenta Nov 18, 2024

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+843

Geometry

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?

AnswerscorrectIy Nov 18, 2024

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+843

Geometry

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 ..

AnswerscorrectIy Nov 18, 2024

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+843

Geometry

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.

AnswerscorrectIy Nov 18, 2024

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+843

Geometry

Point D is the midpoint of median AM of triangle ABC. Point E is the midpoint of AB, and point T is the intersection of BD and ME. Find the area of triangle BEC if [ABC] = 14.

AnswerscorrectIy Nov 18, 2024

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+843

Geometry

A rectangle contains a strip of width $1,$ as shown below. Find the area of the strip.

AnswerscorrectIy Nov 18, 2024

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+843

Geometry

Point $P$ is on side $\overline{AC}$ of triangle $ABC$ such that $\angle APB =\angle PBA$, and $\angle ABC - \angle ACB = 24^\circ$. Find $\angle PBC$ in degrees.

AnswerscorrectIy Nov 18, 2024

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+1439

Counting

In how many ways can you distribute $8$ indistinguishable balls among $5$ distinguishable boxes, if at least three of the boxes must be empty?

kittykat Nov 18, 2024

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