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Math
Since I cannot see the graph, I will provide some basic discussion of this topic to help get you started. Finding the equation of a particular graph depends on what the graph is of. Your graph may...
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Math
A relation is a set of ordered pairs. A relation can be finite (the set has a distinct number of members) or infinite. The set consists of pairs of objects (typically numbers) which are ordered;...
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Math
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). (Stover, Christopher and Weisstein,...
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Math
Making a histogram is similar to making a bar chart in terms of how it looks. However, in a histogram, numbers are grouped into ranges to show the distribution of frequencies. Histograms divide...
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Math
Unit elastic is a scenario or situation in which two variables are related in such a manner that any change in one variable causes an equally proportional change in the other variable. This is...
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Math
Hello! Our task is to evaluate indefinite integral `int ( dt ) / ( 1 + e^t ) . ` To do this, use simple substitution, namely `e^t = u . ` Then `du = e^t dt = u dt , ` or `dt = ( du ) / ( u ) . `...
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Math
Finding antiderivatives is a process that, as the name suggest, is the reverse of that of taking a derivative. It is the same thing as finding an indefinite integral of a function. The common...
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Math
Compound inequalities are two inequalities linked with the words "and" or "or." The possibilities include the following: a<x and x<b (also written a<x<b)a>x and x>b (also written...
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Math
There are two ways to differentiate this integral. First, we can find the indefinite integral `F ( t ) = int 3 t sin t dt ` in elementary functions. Then, using the fact that the definite integral...
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Math
A break-even analysis is an important component of a company's financials because it articulates the level of sales a given product or service must reach in order to cover the initial outlay in...
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Math
To factor an expression, you break it down into factors. The first step should be to identify any common factors. For example, consider the following expression: 6y² + 18y Here 6y² and 18y share...
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Math
Fixed daily cost `C_0 ` (in dollars) means that even to start production, the manufacturer has to pay or spend this amount of money. For example, they need to commute, to pay salary for their...
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Math
The definite integral `int_0^1 t e^(-t^2/2) dt` has to be determined. Let `f = -t^2/2` , `(df)/(dt) = (-2t)/2 = -t` or `df = -t dt` When t = 0, f = 0 and when t = 1, f = -1/2 Substituting t with f...
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Math
We are given a sample mean `bar(x)=150` with a sample size n=200. We are also given a standard deviation: if it is the population standard deviation we have `sigma=30` or it is the sample standard...
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Math
Hello! The position of the center of mass may depend not only on the region's shape but also on its density. The density may vary from point to point and in general may even have point masses (some...
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Math
The mutiplicity of an element in a multiset is the number of times it appears in the multiset. This can be determined in many ways for different cases. If the prime factorization of a number N is...
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Math
We are asked to find the solutions to the following system: `ax+by=a^2+2ab-b^2` `bx+ay=a^2+b^2` Note that a and b are parameters (once chosen they remain fixed) so this is a linear system of...
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Math
A sinusoid can be modeled with a sine curve with possible transformations. A general formula is `h(t)=asin(b(x-h))+k` where h(t) represents the height above 0 at time t, a is the amplitude of the...
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Math
A tire with a nail in it is rotating at one revolution per second and has a radius of 20cm. At time t=0 the nail is on the ground, and we are asked to find a model for the height above the ground...
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Math
The slope-intercept form is given as y = mx + b, where m is the slope, and b is the intercept. The slope is a measure of the steepness of a line and can be calculated as the ratio of change in y to...
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Math
The population of a country in the year 2000 was 6.49 million. It is projected to increase to 16 million by the year 2097. The population of the country after t years can be expressed by the...
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Math
We are told that a drug decays in a person's system according to the decay model `A=A_0e^(kt)` . Here A is the amount of the drug present after t hours, `A_0` (read A nought) is the initial amount...
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Math
If a square or triangle has been drawn on graph paper, you can work out its area just by counting the number of squares. For example, if a square is formed of 25 1cm squares, the area of the square...
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Math
We begin with two bags: bag A contains three bottles of aspirin (3a) and two bottles of thyroid tablets (2t), while the second bag, B, contains two bottles of aspirin (2a), three bottles of thyroid...
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Math
The slope-intercept form refers to the equation of a line. This is the most popular form of the equation of a line. It can be given as: y = mx + b where the slope of the line is given by the symbol...
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Math
Hello! Denote the event "the legislation will pass the House" as `H ` and the event "the legislation will pass the Senate" as `S . ` Then the event in question, "the legislation will pass both the...
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Math
Essentially two polynomials are equivalent if they yield the same output for every input. We can create equivalent expressions by performing logical operations. We are given the expression...
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Math
A mixed number is a number with a proper fraction associated with it. Some examples can be 1 1/2, 3 5/8, 2 7/9, etc. A fraction is generally written in the form a/b where b is not equal to 0. Some...
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Math
Knowing and understanding the contribution margin is vital to establishing a viable path to corporate profitability. First, however, it's important to define exactly what the contribution margin...
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Math
The first step in graphing fractions is to reduce them to their lowest term. For example, consider the coordinate pair (2, 2/4). Here you would want to reduce the fraction to 1/2. Now the...
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Math
f(x)=sin(x); g(x)=sin(x-pi) The graph of g(x) is a translation of the graph of f(x) pi units to the right. This is also described as a phase shift of pi units to the right. The amplitude, period,...
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Math
The predicted yield of the reaction is 21 g but only 3.8 g is recovered. This gives the percentage yield as `(3.8/21)*100 % ~~ 18.095 %` The percentage yield is approximately 18.095%
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Math
The period of a function f(x) = sin x is `2*pi` For sin(x/4) the period is 2*pi*4 = `8*pi` The period of the function `y = sin (x/4)` is `8*pi`
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Math
A complex number a+b*i can be expressed in the polar form as `A*e^(i*theta) ` where `A = sqrt(a^2 + b^2)` and `theta = tan^-1(b/a)` For the complex number 4 + 3*i, A = `sqrt(4^2 + 3^2)` = `sqrt(16...
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Math
The center of mass of the rectangle (1,-3) (4,-3) (4,5) (1,5) has to be determined. The diagram of the rectangle is: The coordinates of the center of mass of the rectangle is given by: `M_x =...
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Math
In lowest terms, 45/30 = 3/2 (not 9/16) because `45/30=(15*3)/(15*2)`
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Math
R and S are relations on set A = {1,2,3,4}, defined as
R = {(1,2},(1,3),(2,3),(2,4),(3,1)}
S = {(2,1},(3,1),(3,2),(4,2)}
Find ` S o R` , `R o S` , `R^-1` , `S^-1` , where `o` means composition.
The relation R = {(1,2),(1,3),(2,3),(2,4),(3,1)} and S = {(2,1),(3,1),(3,2),(4,2)} `SoR` = {(1,1), (2,1), (2,2)} `RoS` = {(2,2),(2,3),(3,2),(3,3),(3,4)} `R^-1` = {(2,1),(3,1),(3,2),(4,2),(1,3)}...
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Math
The SI unit for density is `(kg)/(m^3)` , so we need to convert grams to kilograms, and we already have cubic meters. `6.4x10^5` grams is equivalent to `6.4x10^2` kilograms,as 1000 grams is...
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Math
To find an equation of direct variation, we need the line that goes through the point (-6,1) and the origin. This will get a line of the form y=mx where m is the slope of the line. To find the...
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Math
The line r= (-8,-6,-1) + s(2,2,1) intersects the xz and yz coordinate planes at the point A and B. As r= (-8,-6,-1) + s(2,2,1) intersects the xz plane at A, we have y=0 At A, -6 + 2s = 0 => s =...
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Math
You need to find the resultant vector bar (AB) such that: `bar (AB) = <2 + 1, 3 + 1> => bar (AB) = <3,4>` You need to evaluate the length of diagonal `AB` , hence, you need to...
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Math
The eccentricity of a hyperbola is defined as below. If disance from center to the vertex is a and distance from center to the focus is c, then, eccentricity `= e = c/a` But for a hyperbola, `c...
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Math
First, calculate the amount of the discount: Discount is 22.4% of $173.14. Convert 22.4% into decimal by multipying by 0.01: 22.4% * 0.01 = 0.224 Then the discount equals 0.224*$173.14 =...
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Math
Since each package contains 8 buns, then if you buy n packages, you are able to serve `8n` hamburgers. In addition, serving 14 people two hamburgers each means that you are serving `14 times 2 =...
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Math
The price at which the cameras are bought is $1800 for a dozen. This gives the price of each camera as 1800/12 = $150 He sells them at a profit of $36 per camera. The selling price is $150 + $36 =...
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Math
1) Views by the end of June 2nd = 37 Over each of the next 9 days , the total number of views increases by 35% Therefore views by the end of 11th June = 37*(1.35)^9 = 551 2) Since there were 855...
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Math
A population of E.coli bacteria doubles every 15 minutes. The initial population of the bacteria is 500. An exponential model does not imply the use of 'e'. The general formula for exponential...
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Math
Cody can choose to work any number of hours on each job as long as those hours allow him to earn $693 or more. `(7 $/(hr))(x (hr))+(9 $/(hr))(y (hr))>=$693` or `7x+9y>=693` (1) Find the x...
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Math
We know that the deck of regular play card has 52 cards. These cards are: 1, 2, 3, 4, 5, 6, 7, 8,9, 10, J, Q, and K We have 4 sets of each. Now let us determine the odd numbers in the deck:...
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Math
So we take the derivative of both sides with respect to q and assuming that is a constant we get `0 = -1/p^2 (dp)/(dq) - 1/q^2 ` Solving for `(dp)/(dq)` we get `(dp)/(dq) = - p^2/q^2` So, we have...