Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

learn more… | top users | synonyms (2)

4
votes
3answers
44 views

Inequalities and rearrangements

Some children are arranged in two rows, so that each child in the front row is taller than the child behind him in the back row. The children are now rearranged in increasing order in each row. Show ...
-3
votes
2answers
59 views

Use substitution $x=a\sec(\theta)$ to show that [closed]

Use substitution $x=a\sec(\theta)$ to show that $$\displaystyle\int_{a\sqrt 2}^{2a} \frac{dx}{x^3\sqrt{x^2-a^2}} = \frac{3 \sqrt 3 + \pi - 6}{24 a^3}$$ Need help! Could someone show me the working ...
0
votes
4answers
59 views

Given $x^2 + y^2 = 34xy$, show that $\log\left(\frac{x+y}{6}\right)= \frac{\log x + \log y}{2}$

If $x^2 + y^2 = 34xy$, show that $$\log\left(\frac{x+y}6\right)= \frac{\log x + \log y}{2}.$$ I tried to put log into the first equation, but I have no idea about how the $34$ being simplified in the ...
0
votes
2answers
31 views

Simplification of algebraic expression

I have got a big algebraic expression for function $\vec{a} = \vec{a}(s)$, that i simplified to the form $$\frac{d \vec{a}}{ds} + \frac{a - \sin a}{a^3} (\vec{a} (\vec{a} \cdot \frac{d \vec{a}}{ds}) - ...
0
votes
2answers
35 views

Average speed problem

"I take a journey and, due to heavy traffic, crawl along the first half of the complete distance of my journey at an average speed of $10$ mph. How fast would I have to travel over the second half of ...
0
votes
0answers
50 views

Can you help me with creating a formula [closed]

I make aluminium windows and doors and I would like to create a formula based on an overall width. The scenario is: There are 2 fixed panels of equal length. What are their lengths' as an equation? ...
1
vote
2answers
29 views

How to solve for $t$ in the equation $m=-70/(t-25)$?

In the equation $$m= -\frac{70}{t-25}$$ How would I solve the equation for $t$ so that I get t equal to something $m$? I've tried multiplying both sides by $t-25$, but that just leaves me with ...
2
votes
3answers
37 views

Values of $p$ for which quadratic possess at least one positive root.

For what values of $p$ would the equation $x^2+2(p-1)x+(p+5)=0,\ \ \{x,p\}\in \mathbb{R}$ possess at least one positive root ? I tried $$[2(p-1)]^{2}-4(p+5)\geq 0\\~\\ \implies p\geq 4 \cup p\leq ...
1
vote
3answers
74 views

simplify and evaluate $\frac{\tan80^\circ-\tan20^\circ}{1+\tan80^\circ\tan20^\circ}$ [closed]

How do you simplify and evaluate $\dfrac{\tan80^\circ-\tan20^\circ}{1+\tan80^\circ\tan20^\circ}$? What is the problem asking?
-7
votes
3answers
57 views

Find exact values of $\tan(105^\circ)$ and $\tan(11\pi/12)$ without calculator [closed]

How do you find the exact values of the following without using a calculator? $$\tan(105^\circ) \qquad \tan(11\pi/12)$$
0
votes
6answers
102 views

Factoring $ x^2 + x +1 > 0$ from Spivak Calculus exercise

Hi!! I found me in trouble when I saw the solution of a simple inequality, that can be found at the end of the first chapter, that is the exercise 4 - (viii): $x^2+x+1 > 0$. Very easy to solve I ...
1
vote
3answers
33 views

Area/Sector of a circle: A cow is tethered by a 100-ft rope to the inside corner of an L-shaped building

A cow is tethered by a $100ft$ rope to the inside corner of an L-shaped building, as shown in the figure. Find the area that the cow can graze. (Let $a = 30 ft$, $b = 60 ft$, $c = 100 ft$, $d = 70 ...
2
votes
3answers
73 views

When is $\tan(a+b)$ undefined? [closed]

For what values of $a$ and $b$ is $\tan(a+b)$ undefined? What is the relationship between $a$ and $b$ when it is undefined? What about for $\tan(a-b)$?
1
vote
3answers
58 views

How many possible guesses?

A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $ 1$ to $ 9999$ inclusive. The contestant wins the prizes by correctly guessing the ...
0
votes
2answers
35 views

Co-ordinate geometry involving straight lines $7x-y-32 = 0$ and $3y-2x+1=0$.

Let $P$ be the point of intersection of the lines $7x-y-32=0$ and $3y-2x+1=0$. Lines are drawn through $P$ making intercepts of equal magnitude on the co-ordinate axes. Find the equation of these ...
2
votes
1answer
30 views

Find $p$ and $q$ in $x^2-px+q=0$

If $p$ and $q$ are the roots of the equation $x^2-px+q=0,\ \{x,p,q\}\in\mathbb{R} $, then find $p$ and $q$. I tried sum and product of the roots formula and got , $$p+q=p \\pq=q$$ I found $q=0$ ...
5
votes
1answer
65 views

How many ways are there to shake hands?

In a group of $9$ people, each person shakes hands with exactly $2$ of the other people from the group. Let $X$ be the number of possible ways to perform these handshakes. Take $2$ handshake ...
3
votes
4answers
109 views

Find the matrix $\mathbf{A}$ if $A\binom{7}{-1} = \binom{6}{2}.$

Find the $2\times2$ matrix $A$ where $A^2=A$ and $$A\begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.$$ I tried plugging in: $A= ...
0
votes
3answers
55 views

Let $x_n>a$ for all indices $n$, and $x_n \rightarrow b$. Prove $b \ge a$.

My question then is: Suppose $x_n>a$ for all indices $n$, and $lim x_n=b$. Prove that $b \ge a$. My attempt: I am not going through contradiction-in-conclusion method (i.e. suppose $b<a$ since ...
0
votes
1answer
41 views

Different answers of a quadratic equation.

given $4x^2−4x-5=0$ we all know the solution but what my teacher showed me is different after we get the \begin{align*} x & = \frac{4 \pm \sqrt{96}}{8}\\ x & = \frac{4 \pm \sqrt{4 \cdot ...
1
vote
4answers
132 views

Is 4th root of $-1$ the same as $i^2$?

I am using mathway to check my algebra problems and tried entering in the 4th root of $-1$ (or $(-1)^{1/4}$. I get the same term back, but I thought that since the square root of $-1$ is $i$, that ...
6
votes
2answers
41 views

Sujection, finite set, $|X| \le n$? [closed]

Suppose that $\{1, 2, \dots, n\} \to X$ is a surjection. How do I show that $X$ is a finite set and that $|X| \le n$?
-5
votes
8answers
62 views

Find the smallest integer $n$ [closed]

Find the smallest integer $n$, such that $$n\left ( \sqrt{101}-10 \right )> 1$$
4
votes
2answers
100 views

how to rationalize $\frac {x-8}{\sqrt[3]{x}-2}$

In order to resolve a limit, I need to rationalize $\frac {x-8}{\sqrt[3]{x}-2}$. I tried multiplying it by $\sqrt[3]{x^3}$ or $\sqrt[3]{x^2}$ but with no much success. It seems that I can't use ...
0
votes
0answers
38 views

Newton's Law of Cooling example: roasted turkey

Need help with part B. I keep getting negative answers. A roasted turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 66°F. ...
2
votes
2answers
53 views

Given $|a|$ and $|b|$, what is the smallest value of $|a + b|$?

If a and b are vectors such that $|a| = 7$ and $|b| = 11$, then find the smallest possible value of $|a+b|$. I thought it was $\sqrt{170}$, but its not true. Help! :(
3
votes
1answer
48 views

Rotation of complex numbers in a complex plane. Check my work?

Say that $c_1 = -i$ and $c_2 = 3$. For this problem, let $z_0$ be an arbitrary complex number. We can rotate $z_0$ around $c_1$ by $\pi/4$ counterclockwise to get $z_1$. Next, we canrotate $z_1$ ...
0
votes
1answer
22 views

How to find the initial and the future population based on today's data?

I need help for the part B of the following questions. Here is the question and my work: A certain species of bird was introduced in a certain county $25$ years ago. Biologists observe that the ...
0
votes
1answer
17 views

Solution review

The variables $p,q,r$ and $s$ are correlated with each other with the following relationships $\dfrac{s^{0.5}}{p}=\dfrac{q}{r^2}$ .The ranges of values of $p,q$ and $r$ are respectively: ...
0
votes
3answers
35 views

Use a graphing device to find all solutions of the equation (natural logs) $e^{x^2} − 8 = x^3 − x$

$e^{x^2} − 8 = x^3 − x$ What do I do with the $e^{x^2}$?
5
votes
1answer
48 views

Consider the 1000-element subsets

Consider all 1000-element subsets of the set $A = \{ 1, 2, 3, ... , 2015 \}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, ...
6
votes
2answers
188 views

Find the number of sets of $(a,b,c)$ for $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{29}{72}$

If $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{29}{72},\ \ c<b<a<60,\ \ \{a,b,c\}\in\mathbb{N} $. How many sets of $(a,b,c)$ exists ? Options $a.)\ 3 \quad \quad \quad \quad ...
1
vote
2answers
105 views

Is there a function $f$ such that $f(x) =1$ if $x=p/q$ and $f(x)=0$ otherwise?

Is there a function $f$ such that satisfied : $$f(x) = \begin{cases} 1 \text{ for } x=\frac{p}{q}\\ 0 \text{ for } x \ne \frac{p}{q} \end{cases}$$ and $q\neq 0$ and $x$ is rational number Note : I ...
6
votes
5answers
491 views

Solve the equation. e and natural logs

$$e^x − 6e^{-x} − 1 = 0$$ No idea how to solve this. If someone could show me the first one or two steps to push me in the right direction that would be great.
0
votes
0answers
19 views

Create a recursion here [duplicate]

Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain either exactly two adjacent chairs or no adjacent chairs. I had this question before, but I ...
1
vote
1answer
37 views

Unfairish Probability

Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability $\frac{2}{3}$ and each of the other five sides has probability ...
13
votes
4answers
1k views

Function that looks a lot like exponential, but isn't

I'm looking for a continuous function f(x) with the following properties. I've been playing with exponentials, but that doesn't seem to be the answer, although my high school mathematics is a bit ...
2
votes
2answers
64 views

Number of solutions of $a^{3}+2^{a+1}=a^4$.

Find the number of solutions of the following equation $$a^{3}+2^{a+1}=a^4,\ \ 1\leq a\leq 99,\ \ a\in\mathbb{N}$$. I tried , $$a^{3}+2^{a+1}=a^4\\ 2^{a+1}=a^4-a^{3}\\ 2^{a+1}=a^{3}(a-1)\\ ...
1
vote
3answers
39 views

Logarithmic equation with logarithm in power.

$$x^{\log_{\,3}(3x)}=9$$ I tried to turn the exponential to logarithm form $- \log_{\,x}(9) = \log_{\,x}(3x)$. I also tried using the property $a=\log_{\,b}(b^a)$, but it didn't get me anywhere. I ...
2
votes
4answers
198 views

Nature of the roots of quadratic equation

Here is the problem that I need to prove: If $x$ is real and $\displaystyle{\ p = \frac{3(x^2+1)}{(2x-1)}}$, prove that $\ p^2-3(p+3) \geq 0$ Here is what I did: \begin{align*} p(2x-1)=3(x^2+1) \\ ...
1
vote
4answers
105 views

Median of triangle

I know that a median of a triangle is a line joining one of the vertices to the mid-point of the opposite side. For example, in a triangle OAB, O is the origin, $A$ is the point $(0,6)$ and $B$ is ...
0
votes
0answers
66 views

Why doesn't combinatorics work here?

A while ago I asked one-to-one in combinatorics and then using one-to-one I'll repeat my answer here: There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are ...
-5
votes
2answers
86 views

Simple algebra, but causing arguments in a home school co-op discussion. $6^2/2(3)+4$ [duplicate]

Trying to get some human opinion on this problem. Very simple equation, but it is eliciting quite the controversy. I have executed it in Ruby, Java, C#, Javascript, Python, Perl and Excel with the ...
3
votes
2answers
45 views

Expand the Logarithmic Expression.

Here is the question $\log \left (\sqrt{x^3\sqrt{y^5\sqrt{z}}} \,\right) $ My work: http://i.imgur.com/GBcuzEI.jpg
4
votes
2answers
205 views

Finding a recurrence for a sum

I am trying to implement the following sum using a programming language: $$\sum_{i=1}^N a^i i^r$$ where $N$, $a$ and $r$ are integers. The problem is, I cannot find a suitable way to do this. ...
3
votes
4answers
284 views

why isn't the counting principle giving the right answer?

Note : This is not homework, it is self-study. An employer interviews eight people for four openings in the company. Three of the eight people are women. If all eight are qualified, in how many ...
3
votes
4answers
86 views

Solve for $x$ in $x^3 + (-3)^4 = 17$ [closed]

We have $$x^3 + (-3)^4 = 17$$ Math beginner here! How do you simplify and get $x$? My first time posting so feedback is appreciated!
5
votes
5answers
79 views

Solve $2^{a+3}=4^{a+2}-48,\ a\in \mathbb{R}$

Solve $2^{a+3}=4^{a+2}-48,\ a\in \mathbb{R}$ I tried to simplify it , $2^{a+3}=4^{a+2}-48\\ 2^{a+3}=2^{2(a+2)}-2^4\cdot 3\\ 2^{2a}-2^{a-1}- 3=0\\ $ I don't know how to go from here. This ...
1
vote
1answer
61 views

Solve $n(n+1) \equiv 0 \pmod{1004}$

Solve: $$n(n+1) \equiv 0 \pmod{1004}$$ For the smallest possible $n > 0$. It's either $n \equiv 0$ or $n \equiv -1 \pmod{1004}$. The correct answer is $251$, I'm not sure how though.
0
votes
2answers
44 views

Find value of $t$ between the difference of 3D vectors.

Hint: The distance between $2$ vectors equals the magnitude of their difference. What is the value of $t$ for which the vector $\mathbf v = \begin{pmatrix} 2 \\ -3 \\ -3 \end{pmatrix} + ...