
Math
Hello! We can find the rate at which sand is leaking in volume per time. It is natural to measure time in minutes and volume in cubical inches for this problem. Denote the radius of the cone (and...

Math
There is a story behind this problem. It is said that a famous mathematician by the name of Carl Gauss was punished for poor behavior in class, so he was told to stay after school. As punishment,...

Math
Some of the early computers that used decimal system (either exclusively or in combination with binary) include ENIAC, UNIVAC I and II, and several IBM computers. Some processors today use binary...

Math
The question is a little vague. Ten huskies are to be selected to pull a dogsled. The number of ways in which this can be done is dependent on the total number of huskies from which these ten are...

Math
Hello! The most straightforward way is to use Cramer's Rule. The main determinant of this system is `D = [a,b],[b,a] = a^2b^2 = (ab)(a+b).` If it is nonzero, the system has the only solution....

Math
Hello! I suppose the equation is `1.25+5.25*k` . It may also be `1.25*k+5.25.` a. Then we see that the number `1.25` (the free term) is in dollars (or other currency) and it represents the fixed...

Math
The bell curve essentially "sums up" all the possibilities that can occur given a particular experiment. Especially, in many cases, as the number of times the experiment is repeated, and all...

Math
a) Denote the number of hours it takes Ashlee to repair a DVD by h. Then, the amount of money M Ashlee would charge to repair a DVD, in dollars would be M = 25 + 20h ($25 to inspect and $20 per...

Math
(a) We have to solve the initial value problem given by: `x'=10y` `y'=10x` with initial conditions: `x(0)=3` `y(0)=4` Now we can write the above problem in matrix form as shown:...

Math
According to the variations of parameters method, the formula for the particular solution of the equation of the form `y''+q(x)y' + r(x) y = g(x)` is `Y_P = y_1 int y_2(x)g(x)/(W(y_1, y_2))...

Math
The divergence theorem states that `int_S (F*hatn) dS = int_V (grad*F) dV,` where `S` is a closed surface, `V` is the volume inside it and `F` is a good enough vector field defined inside `S` and...

Math
This function is odd, thus its Fourier expansions contains only `sin(nx)` terms, i.e. `a_n=0, ngt=0.` This expansion has the form `sum_(n=1)^(oo) b_n sin(nx)` where `b_n = 1/pi int_(pi)^(pi)...

Math
We are asked to graph the polar function `r=sqrt(theta), 0<= theta <= 2pi ` , along with its vertical and horizontal tangents. We can graph by plotting points: `theta: ` r:0...

Math
We are given the following coordinate pairs: (17,36), (27,25), (37,20), (47,12), (57,10), (67,7), and (77,5), where the xcoordinate represents the age in years and the ycoordinate the percent of...

Math
The method of disk or washers is based on breaking up the solid obtained by the revolution of the curve into either thin disks or thin washers (rings), and integrating their area to obtain the...

Math
The method of shells is based on breaking up the solid obtained by the revolution of the curve into thin cylindrical shells, and integrating the surface area of the shells to obtain the volume of...

Math
i, j, k are unit vectors. r(u,v)=(b+a*cos(u))*cos(v) i + (b+a*cos(u))*sin(v) j + a*sin(u) k 0<=u,v<=2pi and a=1/4 and b=1 The surface area is: A=double integral of magnitude of (r_u x r_v)...

Math
Please look at the pictures attached.

Math
Denote the given points: First, find the unit vector orthogonal to (for a plane it is a constant vector). An orthogonal vector is Its unit vector is a) Therefore the equation of the plane is or b)...

Math
For each `x,y` the point `(x, y)` is the point where a vector starts and F(x, y) is a pair of its components. For `x=y=0` we have `F(x,y)=vec0` and we cannot draw it with an arrow because it has no...

Math
First octant means `xgt=0, ygt=0, zgt=0.` The formula `y=27/8 x^3` may be written as `y=(3/2 x)^3.` We need the (simple) inverse formulas, too: `x=2/3 root(3)(y), z=1y/2.` Because `zgt=0` we see...

Math
Hello! As I understand, `r = sqrt(x^2 + y^2).` The volume is the integral by dx, dy and dz of the function that is constantly equal to `1.` To set up the integral we need to understand the limits...

Math
How can I find the probability (in fractions)? S={3,5,6,8,9,12,13,14,15,16} A={3,12,5,13}...
I am assuming that we are selecting one item from a sample space of s={3,5,6,8,9,12,13,14,15,16}. We have event spaces: A={3,5,12,13} B={3,6,14,15} C={5,8,9,12,16} We can compute the probabilities...

Math
In the attached image, the boxplot for Region C is both skewed and has an outlier. In statistics, a boxplot is a way to show numerical data in their quartiles. A boxplot consists of a rectangle...

Math
Hello! Although the range is from `1` to `6,` the mean and the distances from the mean are computed using the vertical values (those below `0.2`). They are relative frequencies (probabilities) of...

Math
We are asked to graph `f(x)=(x1)^3(x+4)^2(x2) ` : First we will use only techniques from Algebra. There are a few key attributes of graphs of polynomials we are interested in: end behavior (how...

Math
The negative trend in this scatterplot indicates that the two variables, weight and desired weight change, are negatively correlated. That means as a student's weight goes up, his or her desired...

Math
Well, you can use the information to describe the data. Or, use the data to draw a normal curve, since you know the mean and standard deviation. Now, actually, since these values are suppose to...

Math
We are asked to graph the function `f(x)=1/2(x^22x+15) ` : The graph will be a parabola (f(x) is a quadratic polynomial), opening up (the leading coefficient is positive.) To find the yintercept...

Math
If a computer loses 30% of its value each year, then every year the value of the computer is 70% of its original value. That is, if original value of the computer is $100, then in one year it will...

Math
I will use trigonometric substitution for this integral. First, substitute to eliminate the square root. Then evaluate the integral. Lastly, convert back to the original variable. Please see the...

Math
The volume can be found by using the shell method and multiplying2pi*x and f(x) then integrating over the bounds of x.

Math
This problem is solver by first converting your angle from degrees to radians. Then use the formula for the area of a sector of a circle.

Math
This is an application of the arc length formula. It needs the angle in units of radians. So first convert the angle from degrees to radians. Then apply the arc length formula.

Math
The average rate of change is the slope of the secant line between `x_2` and `x_1` . The slope of the secant line, `m_s`, is found in the attached image.

Math
We are going to make a tangent line of the form eq. (1): y(x)=m*x +b Such that y(x) goes through point (1,2). Therefore y(1)=2 must be satisfied. Here m is the slope that is tangent at point...

Math
Hello! This indefinite integral is simple if we note that `sin(6x) dx = 1/6 d(cos(6x)).` Formally, perform the variable substitution `u = cos(6x),` then `du = 6sin(6x)` and the integral...

Math
This is an integral of the form `oint_C f(z)/(za) dz` , where `f(z)=ze^z` and `a=i` in this case. If this meets the conditions of the Cauchy Integral Theorem we can use the Cauchy Integral...

Math
We must use the following relationship for a complex number `z` . `sin(z)=(e^(iz)e^(iz))/(2i)` `sin(pi/2+i ln(2))=sin(z)=(e^(i(pi/2+i ln(2)))e^(i(pi/2+i ln(2))))/(2i)`...

Math
Let the complex number `z=8` . Rewrite this in polar form as `z=8e^(i(0+2pi*n))` . `z^(1/3)=(8e^(i(0+2pi*n)))^(1/3)` `z^(1/3)=2e^(i(0+(2pi*n)/3))` Here, `theta` becomes 3 distinct angles for all...

Math
The volume between two surfaces can be calculated by `V=int int_D (z_(t o p)z_(b o t)) dA` Where `D` is the area contained by the boundary of the volume projected onto the xyplane. To find `D`...

Math
The complex number `z` has the form `z=x+iy` . In the complex plane where `x` is the real axis and `y` is the imaginary axis we have `x=1` and `y=1` . Find the hypotenuse....

Math
We are asked to determine the value for n such that the indicated sum is correct to 1 decimal place. `S_n=sum_(n=1)^(oo) n/(n^2+n^3)` The series converges (using the comparison test) and the sum is...

Math
a. For the Ratio Test, we need to examine the ratio of (k+1)th coefficient to kth coefficient, here it is `(1/((k+1)!)) /(1/(k!)) = (k!)/((k+1)!) = 1/(k+1).` The limit of this ratio is 0,...

Math
Your description of the median is correct except for one minor, but very important, point. The list must be ordered in order to compute the median. You were given 15,11,12,3,14,17 as your data set....

Math
Hello! I agree with your answer to the part a), the only possible series is `U_n = 4*3^(n1).` The question b) becomes simple if we recall the formula of the sum of `N` terms of a geometric...

Math
Take the derivative of `P(v)` . `P'(v)=17v^2+3*10^3v^2` Set `P'(v)` equal to zero and solve for the critical values. `17v^2+3*10^3v^2=0` `17+3*10^3v^4=0` `3*10^3v^4=17` `v^4=17/3*10^3`...

Math
First, when we complete the square for a quadratic function of the form `f(x)=ax^2+bx+c` we get it into the form `(x+b/2)^2(b/2)^2+c` and then simplify. To do this we must add and subtract (to...

Math
First, know that the vector that is in the direction of the line of intersection `(r)` is the cross product between the normal vectors of the planes since it is perpendicular to each of them. A...

Math
The distance d between two points between `P_1 = (x_1,y_1)` and` P_2 = (x_2,y_2)` is computed using the Pythagorean Theorem. The hypotenuse of a right triangle which has the sides `a = x_2...