All Questions

+189

Algebra

Let f(x) = \left \lfloor \frac{2 - 3x}{3x + 3} \right \rfloor.
Find f(1) + f(2) + f(3) + ... + f(999) + f(1000).

rtsdylifdts 41 mins ago

+189

Algebra

Find the minimum value of \frac{x^2}{x - 1 + x^3} for x > 1

rtsdylifdts 41 mins ago

+189

Algebra

Consider the two expressions $1$ and $\frac{2x+3}{2x+3}.$
a) Show that the two expressions represent equal numbers when $x=10.$
b) Explain why these two expressions do not represent equal numbers when $x=-\dfrac32.$
c) Show that these read more ..

rtsdylifdts 11 hours ago

+189

Algebra

Let $a$ and $b$ be the roots of the quadratic $x^2 - 5x + 3 = 2x^2 + 14x + 8.$ Find the quadratic whose roots are 1/(a + 1) and 1/(b + 1).

rtsdylifdts 11 hours ago

+189

Algebra

Fill in the blanks to make a quadratic whose roots are 5 and -5.
x^2 + ___x + ____

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rtsdylifdts 11 hours ago

+189

Trig

Let x be an acute angle such that \tan x + \sec x = 0. Find $\cos x$.

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rtsdylifdts 11 hours ago

+189

Trig

Let \theta be an acute angle. If $\cos \theta = \frac{1}{4},$ then what is $\sin \theta?$

rtsdylifdts 11 hours ago

+275

Algebra

Simplify \frac{4}{1 - \sqrt[3]{3}} + \sqrt[3]{9}.

Pythagorearn May 5, 2024, 12:57 AM

+275

Algebra

Simplify \dfrac{1}{\sqrt2+sqrt3}+\dfrac{1}{\sqrt2-sqrt3}.

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Pythagorearn May 5, 2024, 12:56 AM

May 4, 2024

+971

Geometry

The tangent to the circumcircle of triangle $WXY$ at $X$ is drawn, and the line through $W$ that is parallel to this tangent intersects $\overline{XY}$ at $Z.$ If $XY = 15$ and $WY = 10,$ find $WX.$

Akhain1 May 4, 2024, 10:01 PM

+971

Algebra

Let
f(x) = \frac{2x + 5}{x - 4} + \frac{x^2 + 1}{4x - 3}.
Find the inverse $f^{-1}(x)$.

Akhain1 May 4, 2024, 10:01 PM

+971

Geometry

In triangle $ABC,$ $\angle C = 90^\circ.$ A semicircle is constructed along side $\overline{AC}$ that is tangent to $\overline{BC}$ and $\overline{AB}.$ If the radius of the semicircle is equal to $3$, and $AB = 1$, then find $\frac{AC}{BC}.$

Akhain1 May 4, 2024, 9:59:06 PM

+20

pls help asap

Let x and y be nonnegative real numbers. if x^2+5y^2=30, what is the max value of xy?

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gaberiellz May 4, 2024, 9:23:35 PM

+971

Algebra

Suppose $a(x) = 2x + 5 - 3x^2 + x^3$ and $b(x) = 4 - x^2 + 4x^4$. Find $(a\circ b)(3) - (b\circ a)(3).$

Akhain1 May 4, 2024, 8:10:30 PM

+971

Algebra

Let x \mathbin{\spadesuit} y = \frac{x^2}{y} for all $x$ and $y$ such that $y\neq 0$. Find all values of $a$ such that $a \mathbin{\spadesuit} (a + 1) = 9$. Write your answer as a list separated by commas.

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Akhain1 May 4, 2024, 8:07:14 PM

+28

Geometry help

Find the area of the triangle. [asy] unitsize(1 cm); pair A, B, C; A = (1,2); B = (0,0); C = (4,0); draw(A--B--C--cycle); label("$\frac{4}{3}$", (A + B)/2, NW, red); label("$2$", (A + C)/2, NE, red); label("$\frac{8}{3}$", (B + C)/2, S, red); [/asy] read more ..

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rtsdyhfdts May 4, 2024, 7:51:25 PM

helpp

For 2021, the amount of U.S. federal tax owed T is a function of taxable income d: read more ..

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userz May 4, 2024, 7:32:46 PM

+971

Counting

Find the number of positive integers that satisfy both the following conditions:
- Each digit is a 1 or a 2 or a 3
- The sum of the digits is 5

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Akhain1 May 4, 2024, 7:10:00 PM

+971

Probability

A member of the junior varsity swim team will be elected as President. Each member is equally likely to be elected. The probability that a sophomore is elected is times the probability that a freshman is elected. Everyone on the junior varsity read more ..

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Akhain1 May 4, 2024, 7:09:45 PM

+189

Geometry

Points $T$ and $U$ lie on a circle centered at $O$, and point $P$ is outside the circle such that $\overline{PT}$ and $\overline{PU}$ are tangent to the circle. If $\angle TOP = 45^{\circ}$, then what is the measure of minor arc $TU$, in deg read more ..

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rtsdylifdts May 4, 2024, 6:28:45 PM

+189

help 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 ..

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rtsdylifdts May 4, 2024, 6:26:33 PM

+189

need help

The diagram below consists of a small square, four equilateral triangles, and a large square. Find the area of the large square.

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rtsdylifdts May 4, 2024, 6:25:39 PM

+28

help pls

Triangle $ABC$ is isosceles with $AB = BC.$ If $AC = 20$ and $[ABC] = 150,$ then find the perimeter of triangle $ABC.$

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rtsdyhfdts May 4, 2024, 6:10:03 PM

+275

Geometry

Semicircles are constructed on AB, AC, and BC. A circle is tangent to all three semicircles. Find the radius of the circle.

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Pythagorearn May 4, 2024, 4:40:09 PM

+275

Geometry

Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19.$ Point $P$ is on segment $AB$ with $AP = 7$, and $Q$ is on segment $CD$ with $CQ = 8$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 25$, find read more ..

Pythagorearn May 4, 2024, 4:39:47 PM

+275

Geometry

A sector of a circle is shown below. The sector has an area of $60 \pi.$ What is the radius of the circle? read more ..

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Pythagorearn May 4, 2024, 4:13:17 PM

+275

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.

Pythagorearn May 4, 2024, 4:12:16 PM

+1145

Geometry

Let P_1 P_2 P_3 \dotsb P_{10} be a regular polygon inscribed in a circle with radius $1.$ Compute
P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1

parmen May 4, 2024, 3:55:37 PM

+1145

Geometry

The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.

parmen May 4, 2024, 3:55:23 PM

+1145

Geometry

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

parmen May 4, 2024, 3:41:38 PM

+1044

Counting

In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings may sit next to each other in the same row, but no child may sit directly in front of their sibling?

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kittykat May 4, 2024, 2:59:36 PM

+1044

Counting

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?

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kittykat May 4, 2024, 2:59:15 PM

+31

help math

A sphere is inscribed in a cone with height 3 and base radius 4. What is the ratio of the volume of the sphere to the volume of the cone?
The original is wrong for some reason.

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Hi69420 May 4, 2024

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need help

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