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.

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1answer
523 views

How do I show that one equation is equivalent to another?

I am being asked to show an equation (a) for thermal expansion is equivalent to another equation (b). Show that the following equation for thermal expansion is equivalent to I have never come ...
0
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1answer
62 views

Limit of a geometric summation

I am trying to solve the following: $$ \lim_{n \to \infty} \sum_{k=1}^{nt} \frac{1}{n} \left( 1 - \frac{1}{n} \right)^{k-1}. $$ I think I was able to get the summation correct (see below): $$ ...
2
votes
1answer
3k views

Learning how to flip equations

I took Algebra and Geometry in high school, never thought I'd use them, then became a programmer. I guess I was wrong. To date, I have the hardest time taking equations and "flipping them," ie: ...
2
votes
1answer
590 views

Find intersection(s) between parametrized parabola and a line

I'm trying to find the value(s) of the parameter $t$ at the intersection point(s) between a 2D general parabola (as a parametric function of $t$) and a line whose equations can be derived from two ...
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1answer
62 views

How to solve for m in this equation?

How can I solve for $m$ in this equation, where $e$ is Euler's number, and $p,k,m \gt 0$, and $p \lt 1$? $$p = \left(1 - e^{\frac{-kn}{m}}\right)^k$$
16
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1answer
62k views

Solving Triangles (finding missing sides/angles given 3 sides/angles)

What is a general procedure for "solving" a triangle—that is, for finding the unknown side lengths and angle measures given three side lengths and/or angle measures?
6
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2answers
294 views

System $a+b+c=4$, $a^2+b^2+c^2=8$. find all possible values for $c$.

$$a+b+c=4$$$$a^2+b^2+c^2=8$$ I'm not sure if my solution is good, since I don't have answers for this problem. Any directions, comments and/or corrections would be appreciated. It's obvious that ...
2
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2answers
175 views

Beginner's algebra (real world application)

This may be too basic a question for this site, in which case I'm sure you'll all let me know. Here is my problem: I have a job worth $\$$40,000 composed of 6 units of x and 3 units of y. I'm trying ...
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2answers
2k views

Adding fractions with addition in the denominator

Let's say I have something like the following: $$\frac{4x+4h}{x+h+1} - \frac{4x}{x+1}.$$ I need to add these fractions together, under a denominator they all share. This would be easy if the ...
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4answers
4k views

Solving a triangle given two side lengths and the measure of a non-included angle

Let's say given an angle A = 46 °, side a = 2.29 and b = 2.71 I figured that the angle B = 58.4 by saying: $$B = \sin^{-1} \left(\frac{ 2.71 \sin{46^{\circ}}}{2.29}\right)=58.4^{\circ}$$ But I ...
2
votes
2answers
181 views

When is $99^{(n+1)}$>$100^n$?

Using logarithms, it doesn't seem all too hard to figure out that $99^{(n+1)}$>$100^n$ when n<457.21 approximately. How does one figure out when $99^{(n+1)}$>$100^n$ without using a single ...
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2answers
2k views

Solving a triangle, given two sides and the measure of the included angle

Let say you have a triangle Angle A = 41 degrees , side b = 3.41 and c = 5.83 can you use pythagoras theorem to find the side a? and how can you find Angle B and C
2
votes
1answer
362 views

Find angles using the Law of Cosines

if you must find the Angle C based on the sides of a = 2, 3 b = 4,6 og c = 5, 9  I have used the formula: $$\cos (C) =\frac{a^2 + b^2-c^2}{2ab}$$ use, but I think i'm doing something wrong: ...
0
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1answer
1k views

Solving triangles and quadratic equations

When calculating the pieces in a triangle with only two sides and an intermediate angle is known, one must solve a quadratic equation. By solving the equation are 2, 1 or 0 solutions, as ...
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2answers
1k views

Find inverse function of exponential function $y=e^x - 2e^{-x}$

How can I find the inverse of $$y=e^x - 2e^{-x}$$ I normally express the function in terms of $x$. But here, $x$ is in the exponent. If I take $\log$ both sides, $\lg{y} = \lg{(e^x - 2e^{-x})}$ ...
3
votes
1answer
87 views

Can $x$ be written as a $\mathbb{Q}[x,y]$-algebraic combination of $x+xy$, $y+xy$, $x^2$, and $y^2$?

I was wondering how to write $x$ as an algebraic combination of $\{x+xy,y+xy,x^2,y^2\}$, with the coefficients $\in \mathbb Q[x,y]$.
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2answers
975 views

11th grade level exponential growth problem?

A certain strain of bacteria that is growing on your kitchen counter doubles every 5 minutes. Assuming you start with only one bacterium, how many bacteria could be present at the end of 96 ...
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2answers
38 views

Solution of a system with exponentials

I want to solve the following system: $$1 + (0.2\pi/W)^{2N} = (1/0.89125)^2$$ and $$ 1 + (0.3\pi/W)^{2N} = (1/0.17783)^2 $$ but i can't see how i can do that without getting to many confusing ...
0
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1answer
59 views

Please tell me if I did this calculus separation pretend algebra stuff all right

Define $D_q=\frac{dD}{d_q}$ Is the following true or have I taken too many liberties with the Leibniz quasi-algebra? ...
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2answers
169 views

How does one define cross product

Cross product has a wide application in many field like in physics.Torque and circular motion are its application. But how does one define cross product and why they defined in that way??
3
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1answer
407 views

Help in understanding the properties of prime numbers

I was reading about hashing. The oldest/standard approach is to use a prime number to produce the hash. At first I couldn't get why use a prime when I came to this Why hash functions use primes: ...
1
vote
1answer
327 views

Simplifying a Multi-Variate Fraction

I am an eighth grader in need of a little assistence. I was given a multi-variate fraction, and was told to simplfy it to lowest terms. On of my fellow classmates that is ahead of me in math, tried ...
4
votes
1answer
93 views

Does the logistic function really uniquely satisfy this?

It is said that the logistic function (denoted $y(u)$ below) is derived from the relation: $$\frac{dy}{du}=y(u)(1-y(u))$$ Does $y(u)=\frac{1}{1+e^{-u}}$ really uniquely satisfy this? I don't see ...
2
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2answers
897 views

Three equations (almost linear), five unknowns, solve for three variables.

This problem doesn't seem to make sense to me. I have the following three equations: $$ S\alpha+1.06\beta + \mathcal{F} = S\\ T\alpha+1.06\beta + \mathcal{F} = T\\ 98\alpha+\beta + 0\mathcal{F} = ...
2
votes
1answer
121 views

Vectors, just show that two points are equal? How many approaches are there?

This is a fairly simple thing to do, but what would be the optimal approach to solve part (c) (parts (a) and (b) are done): In the triangle ABC M is the midpoint on AB. Let OA = $\vec{a}$, OC = ...
1
vote
1answer
154 views

Von Mises width at half height

I'm fitting the following Von Mises type function to some data: $(A/2\pi)e^{k\cos(\theta)}+C$ where A and k are positive. I want to calculate the width at half height from the lowest point of the ...
2
votes
1answer
68 views

Simplify _Elementary Calculus_ section 1.6 problem 33

Once again, I'm trying to simplify an expression from Elementary Calculus with hyperreals. Given that $H$ is infinite, compute the standard part of: $$\frac{\sqrt{H+1}}{\sqrt{2H}+\sqrt{H-1}}$$ The ...
3
votes
1answer
285 views

Contest problem on domain and range of square root function

I have no clue how to do this problem: Let $f(x)=\sqrt{ax^2+bx}$. For how many real values $a$ is there at least one positive real value of $b$ for which the domain of $f$ and the range of $f$ are ...
2
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3answers
80 views

Vectors, find implied dot product?

The task goes as following. The angle between two vectors $\vec{w}$ and $\vec{r}$ is less than 90 degrees. Vector $\vec{w}$ is given by $\vec{w} = \vec{u} + \vec{v}$ where $\vec{u} \parallel ...
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3answers
2k views

How can I calculate the time two people will meet if they are paddling towards each other on a lake?

Here's the question I'm having trouble with. Ken and Kara are 30 miles apart on a calm lake paddling towards each other. Ken paddles at 4 mph, while Kara paddles at 7 mph. HOw long will it take them ...
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2answers
255 views

A line moving along the hypotenuse of a right triangle

ABC is a triangle with sides $AB = 6 m$, $BC = 8m$, and $AC = 10m$. A line $k$ in the plane of the triangle $ABC$ moves along the segment $AC$ at the rate of $1cm$ per sec. The line starts at A and ...
0
votes
2answers
209 views

Is this proof by contradiction using sets correct?

Suppose $A \subseteq B$ and $x \in A $ and $x \notin B $ \ $C$. Prove that $x \in C$. Basically what i need to do is to prove this by contradiction, so what i made was: first of all, by applying ...
2
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1answer
434 views

closed form solution for summation of $\log(i)$

Is there a way to find a closed form solution for: (Note that base is $2$) $\displaystyle\sum_{i=1}^n\log_2(i)$ thanks for any help Can't find a formula for this
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2answers
83 views

Simplify _Elementary Calculus_ section 1.6 problem 25

Let's try this again. We're still on problem 25 in section 1.6 of Elementary Calculus. $$\frac{3-\sqrt{c+2}}{c-7}$$ My first thought is (again) to multiply by $3+\sqrt{c+2}$: ...
0
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1answer
213 views

Stacking cylinders

I have three identical, vertical, cylindrical tanks each with a diameter of 1040mm. They need to be crated. What is the minumum internal area of the rectangular crate required ?
2
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1answer
104 views

Simplify _Elementary Calculus_ section 1.6 problem 25 (incorrect transcription)

Given $c\neq7$ and $st(c)=7$, simplify $$\frac{3-\sqrt{c+2}}{\sqrt{c-7}}$$ My inclination, based on one of the examples in the book, is to multiply by $3+\sqrt{c+2}$, yielding: ...
3
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3answers
2k views

Writing a Polar Equation for the Graph of an Implicit Cartesian Equation

If $(x^2+y^2)^3=4x^2y^2,$ then $r=\sin 2\theta$ for some $\theta$. Using $r^2=x^2+y^2, x=r\cos\theta,y=r\sin\theta$, it's easy to get $r^2=\sin^22\theta$. But I don't know what to do next, since ...
2
votes
1answer
283 views

Suppose $A \subseteq C$ and $B$ and $C$ are disjoint. Prove that $x \in A \rightarrow x \notin B$

Suppose $A \subseteq C$ and $B$ and $C$ are disjoint. Prove that $x \in A \rightarrow x \notin B$. Basically I need to prove this.
3
votes
2answers
960 views

The inequality $b^n - a^n < (b - a)nb^{n-1}$

I'm trying to figure out why $b^n - a^n < (b - a)nb^{n-1}$. Using just algebra, we can calculate $ (b - a)(b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1}) $ $ = (b^n + b^{n-1}a + \ldots + ...
4
votes
4answers
8k views

How to solve a quartic equation?

Could someone please explain how to solve this : $x^4 - 10x^3 + 21x^2 + 40x - 100 = 0$ - not the answer only, but a step-by-step solution. I tried to solve it, with the help of khanacademy, but still ...
2
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1answer
298 views

Word problems - Sum of squares & a strange function

These were two of 20 problems I had to do in a test today that I didn't manage to solve. 1) Find the least $k$ such that $1^2 + 2^2 + 3^2 + 4^2 + \dots + k^2$ is a multiple of 200. 2) ...
2
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2answers
176 views

Queueing Theory: Why does this hold for a M/M/1 queue?

For a M/M/1 queue, calculating the estimated number of jobs $n$ in the queue is given by: $$E[n] = \sum_{i=1}^{\infty} p_i i = \sum_{i=1}^{\infty} \rho^i (1-\rho) i .$$ The final result for a M/M/1 ...
20
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3answers
790 views

A problem about floor function

Here's the problem Suppose that $$ a, b \in \mathbb {R^+},\qquad 0 < a + b < 1 $$ Prove or disprove that $$ \exists n \in \mathbb{Z^+}: \left\{na\right\} + \left\{nb\right\} \ge 1$$ ...
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1answer
1k views

Which function is like the cubic function rotated 90 degrees?

I need something like the cubic function $y=x^3$ rotated 90 degrees. Which function is that?
0
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1answer
142 views

Inequality regarding exponents(similar to Bernoulli's inequality)

I was trying to solve a problem today when I felt that if the following is proved, we are done: $(1+a)^y<(1+ay)$ for $0<y<1$ for all non-zero reals a,y.I am not able to make much progress on ...
11
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1answer
2k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
0
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1answer
107 views

A function with certain properties to determine the price of an item

First off, I'd like you guys to take into consideration that I have no clue what I'm trying to do "mathematically speaking". I am here trying to find help with a function for my website where I sell ...
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2answers
54 views

A question about finding the time difference

Two people $A,~B$ are participating a running race.Initially of course they are both at rest.They then proceed with constant acceleration. $A$ covers the last $\dfrac{1}{4}$ of the distance in $3$ ...
1
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2answers
772 views

How do I find the area of multiple shaded rectangles in a parabola?

It's most easily explained if you look at this image (sorry for the bad quality!): I haven't drawn it too well, the vertex should be at the y-axis. I'm not sure how to find the area of the shaded ...
3
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3answers
502 views

Deriving two properties of clocks

This two properties are given in my module as formulas for clock related problems: $(1)$ If both the hands start moving together from the same position, both the hands will coincide after $ ...