AnswerscorrectIy

198 Questions
17 Answers

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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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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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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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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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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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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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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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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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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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Number Theory

Jon teaches a fourth grade class at an elementary school where class sizes are always at least 10 students and at most 15. One day Jon decides that he wants to arrange the students in their desks in a rectangular grid with no gaps. Unfortunately read more ..

AnswerscorrectIy Oct 29, 2024

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Algebra

Expand (x + 2)(3x^2 + 12) + (x - 2).

AnswerscorrectIy Oct 28, 2024

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Geometry

In triangle $ABC,$ the angle bisector of $\angle BAC$ meets $\overline{BC}$ at $D.$ If $\angle BAC = 60^\circ,$ $\angle CAD = 45^\circ,$ and $AD = 24,$ then find the area of triangle $ABC.$

AnswerscorrectIy Oct 27, 2024

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Geometry

In triangle $ABC,$ $AB = 15,$ $BC = 9,$ and $AC = 10.$ Find the length of the shortest altitude in this triangle.

AnswerscorrectIy Oct 27, 2024

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Geometry

In triangle $ABC,$ $\angle B = 90^\circ.$ Point $X$ is on $\overline{AC}$ such that $\angle BXA = 90^\circ,$ $BC = 15,$ and $CX = 5$. What is $BX$?

AnswerscorrectIy Oct 27, 2024

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Geometry

In triangle $ABC$, point $D$ is on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 45^\circ$, $\angle C = 45^\circ$, and $AC = 12$, then find the area of triangle $ABD$.

AnswerscorrectIy Oct 27, 2024

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Algebra

Find all values of c such that 3(2c + 1) = 28*(3c) - 9 + 45(3c) - 18. If you find more than one value of c, then list your values in increasing order, separated by commas.

AnswerscorrectIy Oct 24, 2024

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Algebra

Compute the sum
\frac{1}{\sqrt{36} + \sqrt{39}} + \frac{1}{\sqrt{39} + \sqrt{40}} + \frac{1}{\sqrt{40} +\sqrt{45}}

AnswerscorrectIy Oct 24, 2024

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Algebra

Let $x$ and $y$ be complex numbers. If $x + y = 2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?

AnswerscorrectIy Oct 24, 2024