Homework Help

# Ask a tutor in *any* subject

and get expert help.

## Answers in a flash

Quality answers—when you need them

## Thousands of answers

Access a vast library of expert answers. To date we've answered 369,667 questions, and new answers are added every day!

## Any subject

From *Hamlet* to the Revolutionary War to biology—we cover it all. Ask a new question or browse the hundreds of thousands already asked by other students.

## Certified experts

Our certified Educators are professors, teachers, and scholars who use their academic expertise and professional teaching experience to tackle your toughest questions.

## Rigorous editing process

No other Q&A service has our quality standards. We pride ourselves on our edited, fact-checked, and original content. Learn about our editing process.

## Explore all 0 questions

Math

### inverse sqrt(1-x^2). 0

Given the function `f(x)=\sqrt{1-x^2}` . We have to find the inverse of this given function i.e. `f^{-1}(x)` . An inverse function (also known as anti function) is defined as a function which can...

Math

### Ali told Greg that 2`^607` - 1 is not divisible by any prime number less than 10. Why is Ali correct?

The prime numbers less than 10 are 2, 3, 5, and 7. It is obvious that `2^607 - 1 ` is not divisible by 2. To prove the same for other primes, use the facts that `( a + b ) mod c = ( a mod c ) + ( b...

Math

### Art told Linsey that in triangle ABC, medians AD and BE intersect at G, and triangle AGE is equilateral. Then cos(C)...

Denote the length `A G ` as `x , ` then by the conditions `G E = A E = C E = x . ` By the median property `G D = 1 / 2 x ` and `G B = 2 x .` By the cosine theorem, `E D^2 = x^2 + 1 / 4 x^2 + 1 / 2...

Math

### Sam told Garth that Isosceles trapezoid ABCD has parallel sides AD and BC, with BC< AD and AB =CD. There is a...

Denote `AD = 2 a ` and `BC = 2 b , ` then the question becomes what is `b / a . ` Also, denote the height of the trapezoid as `h .` Introduce a coordinate system such that `AD ` lies on the x-axis...

Math

### Ron told Elsie that P(x) is a polynomial with rational coefficients such that when P(x) is divided by the polynomial...

It is given that `P ( x ) = q ( x ) ( x^2 + x + 1 ) + ( x + 2 ) = q ( x ) ( x^2 + 1 ) + ( 2x + 1 ) , ` where q and r are polynomials. p and q cannot have degree less than 0, so P cannot have degree...

Math

### Robin asked Rick to suppose that x and y are positive real numbers such that x`^y` = 2`^64` and (log2 x)`^log2 `...

We have two equations for `x ` and `y: ` `x ^ y = 2 ^ 64 ` and `( log _ 2 x ) ^ ( log _ 2 y ) = 2 ^ 27 . ` Let's apply `log _ 2 ` to both sides of them: `y * log _ 2 x = 6 4 , ` `log _ 2 y * log...

Math

### Sam told Marcus that the graph of y = x`^2` + 2x - 15 intersect the x-axis at point A and C and the y-axis at point...

It is simple to find the coordinates of the points A, B, C: `A ( -5 , 0 ) , ` `B ( 0 , -15 ) , ` `C ( 3 , 0 ) . ` Because of this, it is simple to find the lengths of the sides AB, BC, AC: `AB =...

Math

### Given that S is the set of circles in the coordinate plane that are tangent to each of the three circles with...

There are two somewhat different types of tangent circles: they may lie on different sides of their common tangent, or on one side. Because the first and the third circles lie inside the second one...

Math

### Mary told May to let R be the region in the complex plane consisting of all complex numbers Z that can be written as...

The set of all complex numbers of magnitude of at most 1 is he unit circle. Its area is `pi * 1 ^ 2 = pi approx 3 . 1 4 .` Addition of complex numbers is component-wise, so when we add all points...

Math

### Why is the set of all complex numbers of the magnitude of at most 1 a unit circle?

Well, there might be some linguistic difficulty because there is actually one word for "circle as a curve" and "circle as a curve and everything inside it." For the curve, we can also say...

Math

### Show the cases when a = -b and b = - c for your solution for the following problem: a, b, and c are real numbers that...

When analyzing the equality `1 / a + 1 / b + 1 / c = 1 / ( a + b + c ) , ` we noticed that it is equivalent to `( a + b ) ( b + c ) ( a + c ) = 0 ` (provided of course that `a ! = 0 , ` `b ! = 0 ,...

Math

### Please use a diagram to illustrate what is meant by the following: "Consider a section of the pyramid by a vertical...

No problem. I used your image to show where the section resides. Also, I have drawn an image with the section itself. The section of the cube `C ` is a rectangle `Y _ 1 Z _ 1 Z _ 3 Y _ 3 . ` Its...

Math

### How do you write the equation of a parabola that crosses the y-axis at y = -1/2 and the x-axis at (-1,0) and (1,0)?

Suppose that the parabola has the equation of the form `y = a x ^ 2 + b x + c .` For each given point with the coordinates `( x _ k , y _ k ) ` we obtain an equation for `a , ` `b ` and `c ` in the...

Math

### Jim said to Harry that a set of positive integers {a, b, c} satisfies a`^2` + b`^2` = c`^2` and assumes c = b + 1....

Actually, the condition `c = b + 1 ` is sufficient to conclude that `g c d ( b , c ) = 1 , ` the more `g c d ( a , b , c ) = 1 .` Indeed, let some integer `n ` divides both `b ` and `c , ` i.e. `b...

Math

### There are three equations in three unknowns of the form aix + biy + ciz = di (i = 1, 2, 3), where the coefficients...

The equations may be reduced by n! , (n + 4)!, and (n + 7)! , respectively, and the augmented matrix becomes `( ( 1 , n + 1 , ( n + 1 ) ( n + 2 ) , ( n + 1 ) ( n + 2 ) ( n + 3) ), ( 1 , n + 5 , ( n...

Math

### Samuel told Daniel that a, b, and c are real numbers that satisfy 1/a + 1/b + 1/c = [1/(a + b + c)], show that 1/an +...

Determine what the condition `1 / a + 1 / b +1 / c = 1 / ( a + b + c ) ` actually means. Namely, multiply the equation by `a , ` `b , ` `c ` and `a + b + c :` `( b c + a c + a b ) ( a + b + c ) =...

Math

### A right pyramid with apex A over the center of a square base w=2 and let the pyramid height be h = 2. Let C be a...

Consider a section of the pyramid by a vertical plane that goes through a diagonal of its base. It forms an equilateral triangle of the height `h ` and base, say, `2 a , ` in which a rectangle of...

Math

### f(x)=(tan x)^1/x use logarithmic differentiation to find f'(x)

Given `f(x)=(tanx)^{\frac{1}{x}}` . We have to find `f'(x)` using logarithmic differentiation. Taking natural log on both sides of the equation we get: `lnf(x) = ln((tanx)^\frac{1}{x})` Applying...

Math

### y=x^k. k is a constant. Find the area under the curve from x=0 to x=1.

We have to calculate the area under the curve `y=x^k.k` (where `k` is a constant) from `x=0` to `x=1` (upper and lower limit). We can use the integration process to find the area under the curve....

Math

### Quiz has two multiple choice questions with four possible answers. What is the probability of getting both questions...

The multiple-choice questions aren't connected to one another, meaning the student could get either, both, or neither right. In a situation where you've got two independent events, the probability...

Math

### Sam told Vince that triangle ABC has sides of lengths AB = 16, BC = 12, and CA = 20. Three circles, all of radius 10,...

The total area covered by the three circles is 942.48. The formula for calculating this is A = π r² (314.16) multiplied by three. However, the wording of the question is ambiguous. Since the...

Math

### In triangle ABC, angle ACB = 60 degrees and angle BAC = 75 degrees. AD is perpendicular to BC at D and BE is...

`A D ` and `B E ` are both heights of the triangle `A B C , ` so their point of intersection `H ` is the orthocenter. The third height goes through the same point `H , ` i.e., the line `C H ` is...

Math

### John told Carl that triangle ABC is an isosceles right triangle. Point P is inside triangle ABC such that PC = 13 ,...

The answer is: it is impossible with these numbers. Indeed, let the right angle be B. Denote the leg length as a and place the triangle inside a coordinate system such that B(0, 0), A(a, 0) and...

Math

### In triangle ABC, AB = AC, D is the midpoint of BC, and E is the midpoint of AD. BE and AC intersect at point F. Prove...

Let's use the coordinate method. The origin will be `D , ` the x-axis will be `D C ` and the y-axis will be `D A .` Let `D C = a , ` `D A = b , ` then `A ( 0 , b ) , ` `B ( -a , 0 ) , ` `C ( a , 0...

Math

### find the equation of the tanget line to the function f(x) = -x3 + x at the point where x = 2.

The equation of the line tangent to `f(x)=-x^3+x` when x = 2 is y = -11x + 16. To find the equation of a line, we need the slope of the line and a point. Since the tangent line intersects the curve...

Math

### Is x =33/7 one of the elements of x in all triples of real numbers x, y, and z that satisfy the system of equations x...

Given the system of equations: `x+xy+xyz=33` ------>(1) `y+yz+yzx=60` ------>(2) `z+zx+zxy=42` ------>(3) We have to find whether `x=33/7` is one element of all triples of real numbers x,...

Math

### Is x=5.5 one element of all triples of real numbers x, y, and z that satisfy the system of equations x + xy +xyz =...

Given the system of equations: `x+xy``+xyz=33`------>(1) `y+yz+yzx=60` ------>(2) `z+zx+zxy = 42` ------>(3) We have to find whether `x=5.5` is one element of all triples of real numbers...

Math

### Judy in return asked Jane to consider a fixed circle of diameter d, and an arbitrary point P outside the circle and...

"Within distance d from the circle" probably means not "the distance from the circle is d" but "the distance from the circle is less than d." Denote this distance as `a. ` Then the solution...

Math

### Mary said to Ken given any nonempty set of numbers X ={x1, x2,...xk}, (where 1 to k are subscripts).Define...

The numbers that form all sums in question are 1, 2, 3, 4, 5, 6, 7, 8, and their squares are 1, 4, 9, 16, 25, 36, 49, 64. Determine the sums of all numbers for `4 lt= n lt= 8 : ` 10, 15, 21, 28,...

Math

### Jane asked Judy to consider a fixed circle of diameter d and an arbitrary point P outside the circle and within...

Denote the center of the circle as `O , ` then denote `O R = O Q = r , ` then `O P = 3 r .` Also denote `P Q = Q R = x` and `/_ R Q O ` as `alpha .` Then apply the Cosine law to the triangles `O Q...

Math

### Find all triples of real numbers x, y, and z that satisfy the system of equations x + xy +xyz = 33; y + yz +yzx =...

I suppose the third equation should be `z + zx + zxy = 42 ` (zx , not yz). Subtract the first equation from the second, then subtract the third equation from the second and express `z ` from the...

Math

### Three distinct points are chosen at random on a circle (each point on the circle has the same chance of being...

Let the circle be a unit circle centered at the origin: this does not affect the probability. Moreover, the position of the first point does not change the probability, so we may even suppose that...

Math

### Samuel told Ann that in right triangle ABC, angle BAC = 90 degrees and P and Q are points on AB and AC such that MP...

We are given right triangle ABC with right angle at A. We assume M to be a point on BC (the hypotenuse) with BM = MC (thus, M is the midpoint of the hypotenuse). Locate points P on AB and Q on AC...

Math

### In a right triangle ABC, angle C = 90 degrees and angle A = 30 degrees. AE is the angular bisector of angle A and...

Denote the side BC as `x, ` then `AB = 2x ` and `AC = x sqrt(3) . ` The area in question is `A = sqrt(3) / 2 x^2 .` Also denote `CE = y, ` then `BE = x - y .` By the Pythagorean theorem `3x^2 +...

Math

### Point P is inside square ABCD such that PA = 5, PB = 8, and PC = 13. Find the area.

First, we need to find the area of the square. Denote the side length as `a , ` the distance from P to AB as `x ` , and the distance from P to AD as `y . ` Then we have three equations for three...

Math

### One can compute the number of ordered pairs of positive integers (m, n) such that m ≤ n`^2 ` and m + n ≤ 53. How can...

It is evident that `1 lt= n lt= 52 . ` There are two subsets of `n , ` namely such that `n^2 lt 53 ` and `n^2 gt 53 , ` which are the same as `1 lt= n lt= 7 ` and `8 lt= n lt= 52 .` For `n ` from...

Math

### The base of a pyramid is a regular decagon of side length 2 and area B. The height of the pyramid is 1. The total...

Consider two adjacent vertices of the polygon, `A_1 ` and `A_2 , ` and its center `O .` Clearly `O A_1 = O A_2 ` and the median `O C ` is also the height and the bisector of the triangle `A_1 O...

Math

### Let r, s, t be the roots of p(x)=x^3-8x+1. Therefore, compute ((r-s)^2)((s-t)^2)((t-r)^2).

We know that `( x - r ) ( x - s ) ( x - t ) = x^3 - 8x + 1 ` for any `x . ` This means that `rst = -1 , ` `r + s + t = 0 , ` `rs + st + tr = -8 . ` Differentiate the first equality w.r.t. `x:`...

Math

### Integrate the indefinite integral. \int (3x^(3)-x^(2)+6x-4)/((x^(2)+1)(x^(2)+2))dx

First, we should represent the function under the integral as a sum of partial fractions. Then it will integrate easily using arctangent and/or logarithm. The decomposition should exist in the...

Math

### Compute the positive real value of x such that `root(3)(20x +22)` - `root(3)(20x -22)` = 2.

To begin, let's move the second root to the right side and raise the equation to the third power: `20x + 22 = 8 + 12 ( 20x - 22 )^( 1 / 3 ) + 6 ( 20x - 22 )^( 2 / 3 ) ,` or `3 ( 20x - 22 )^( 2 / 3...

Math

### John told Mike that a four-digit positive integer is called consecucute if and only if the two-digit number formed by...

Let the two rightmost digits will be `a ` and `b , ` then `0 lt= a , b lt= 9 . ` The two-digit number composed of the two leftmost digits is equal to `( bar ( a b ) - 1 ) ,` and it must be less...

Math

### Solve for x=0 in the following equation: F(x)=67sin(12(x+o.o223))+70.

We are finding F(x) if x=0, so substitute 0 for x in the equation and start within the parentheses (PEMDAS). F(0)=67sin(12(0.223))+70 Finish the parenthetical, so multiply 12*0.223...

Math

### Compute the hundredth digit of (11)`^2022`.

To find the hundredth digit of a number, it is sufficient to compute this number modulo 1000. Because 11 and 1000 are coprime, Euler's totient theorem may be applied, so`11^( phi ( 1000 ) ) = 1...

Math

### What is 111 + 11? How do I solve this problem?

The result of the operation "111+11" is 122. You can find the solution in a number of ways, and the correct method depends on what makes the most sense to you. Some people find it easier to break...

Math

### How would you solve this example problem? Let K be a point in triangle ABC.AK, BK, and CK cut sides BC, CA, and AB at...

We'll use some area considerations. Let's drop the perpendiculars `C C_2 ` and `K C_3 ` to the side `A B ` (look at the attached picture). Then the triangles `C C_2 C_1 ` and `K C_3 C_1 ` are...

Math

### What is algebra?

Algebra is a branch of mathematics that deals with variables (like x, y, and z) that represent unknown quantities. It builds on the fundamental arithmetic skills of addition, subtraction,...

Math

### How do you solve the equation 4x+3y+12z+22=21x+30y+100z-200?

You will be able to solve for one variable in terms of another, but without a second equation to create a system of equations, you won't be able to calculate the actual numerical value of any...

Math

### Given the positive integers p, q, r, s, t, u, v, if the sum of the values of each group of three consecutive letters...

In this question, we have seven positive integers (p, q, r, s, t, u, and v). It's said that the sum of each group of three consecutive letters equals 35. Also, q+u=15. So, by inference, we can...

Math

### How many numbers between 100 and 999 are there in which the middle digit equals the sum of the other two digits?

For the number 385, the middle digit equals the sum of the other two digits (8=3+5). To find numbers between 100 and 999 with this same property, we perform the following calculations. Note that...

Math

### Let a1 = 2021 and for n `>=` 1 let an + 1 =`sqrt(4 +an ` Then a5 can be written as `sqrt(m/2 + sqrt(n) /2` +...

Given that `a_1=2021` for `n>=1` and let `a_{n+1}=\sqrt{4+a_n}` ----->(1) then, `a_5` can be written as `a_5=\sqrt{\frac{m}{2}+\frac{\sqrt{n}}{2}}+\sqrt{\frac{m}{2}-\frac{\sqrt{n}}{2}}`...

*Showing 1-50 of 44459*