MATHEMATICS COLLEGE

The length of a rectangle is represented by (6x − 2), and the width is represented by (x − 1). Which expression best represents the area of the rectangle? A) 6x2 + 8x + 2 B) − 8x + 2 C) 6x2 − 8x + 2 D) 6x2 − 8x − 2

Answers

Answer 1

Answer:

Answer: Area = $6x^(2)-8x+2$

Step-by-step explanation:

The formula for calculating the area of a rectangle is given by :

Area = Length x width

length = $(6x-2)$

Width = $(x-1)$

Therefore :

Area = $(6x-2)(x-1)$

Expanding the equation , we have

Area = $6x(x-1) -2(x-1)$

Area = $6x^(2)-6x$ $-2x + 2$

Area = $6x^(2)-8x+2$

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Tanya enters a raffle at the local fair, and is wondering what her chances of winning are. If her probability of winning can be modeled by a beta distribution with α = 5 and β = 2, what is the probability that she has at most a 10% chance of winning?

Answers

Answer:

$P(X<0.1)= 5.5x10^(-5)$

Step-by-step explanation:

Previous concepts

Beta distribution is defined as "a continuous density function defined on the interval [0, 1] and present two parameters positive, denoted by α and β, both parameters control the shape. "

The probability function for the beta distribution is given by:

$P(X)= (x^(\alpha-1) (1-x)^(\beta -1))/(B(\alpha,\beta))$

Where B represent the beta function defined as:

$B(\alpha,\beta)= (\Gamma(\alpha)\Gamma(\beta))/(\Gamma(\alpha+\beta))$

Solution to the problem

For our case our random variable is given by:

$X \sim \beta (\alpha=5, \beta =2)$

We can use the following R code to plot the distribution for this case:

> x=seq(0,1,0.01)

> plot(x,dbeta(x,5,2),main = "Beta distribution a=5, b=2",ylab = "Probability")

And we got as the result the figure attached.

And for this case we want this probability, since we want the probability that she has at most 10% or 0.1 change of winning:

$P(X<0.1)$

And we can find this probability with the following R code:

> pbeta(0.1,5,2)

[1] 5.5e-05

And we got then this : $P(X<0.1)= 5.5x10^(-5)$

Solve the following system of equations using the elimination method.7x + 20y = 14
2x – 10y = 4
Question 18 options:

A)

(3,1)

B)

(2,0)

C)

(4,–5)

D)

(–3,4)

Answers

Answer:

opt B (2,0)

Step-by-step explanation:

pls rate

Answer:

Step-by-step explanation:

A submarine was 350 feet below sea level. Then, it rose 75 feet, before diving back down another 100 feet. What is the current elevation of the submarine?

Answers

The submarine would be 375 feet below sea level.

Since this is talking about BELOW sea level, you would subtract 75 from 350 instead of adding, which would give you 275. Then, you would add 100 to that instead of subtracting, which gives to 375.

Name/ Uid:1. In this problem, try to write the equations of the given surface in the specified coordinates.(a) Write an equation for the sphere of radius 5 centered at the origin incylindricalcoordinates.(b) Write an equation for a cylinder of radius 1 centered at the origin and running parallel to thez-axis inspherical coordinates.

Answers

To find:

(a) Equation for the sphere of radius 5 centered at the origin in cylindrical coordinates

(b) Equation for a cylinder of radius 1 centered at the origin and running parallel to the z-axis in spherical coordinates

Solution:

(a) The equation of a sphere with center at (a, b, c) & having a radius 'p' is given in cartesian coordinates as:

$(x-a)^(2)+(y-b)^(2)+(z-c)^(2)=p^(2)$

Here, it is given that the center of the sphere is at origin, i.e., at (0,0,0) & radius of the sphere is 5. That is, here we have,

$a=b=c=0,p=5$

That is, the equation of the sphere in cartesian coordinates is,

$(x-0)^(2)+(y-0)^(2)+(z-0)^(2)=5^(2)$

$\Rightarrow x^(2)+y^(2)+z^(2)=25$

Now, the cylindrical coordinate system is represented by $(r, \theta,z)$

The relation between cartesian and cylindrical coordinates is given by,

$x=rcos\theta,y=rsin\theta,z=z$

$r^(2)=x^(2)+y^(2),tan\theta=(y)/(x),z=z$

Thus, the obtained equation of the sphere in cartesian coordinates can be rewritten in cylindrical coordinates as,

$r^(2)+z^(2)=25$

This is the required equation of the given sphere in cylindrical coordinates.

(b) A cylinder is defined by the circle that gives the top and bottom faces or alternatively, the cross section, & it's axis. A cylinder running parallel to the z-axis has an axis that is parallel to the z-axis. The equation of such a cylinder is given by the equation of the circle of cross-section with the assumption that a point in 3 dimension lying on the cylinder has 'x' & 'y' values satisfying the equation of the circle & that 'z' can be any value.

That is, in cartesian coordinates, the equation of a cylinder running parallel to the z-axis having radius 'p' with center at (a, b) is given by,

$(x-a)^(2)+(y-b)^(2)=p^(2)$

Here, it is given that the center is at origin & radius is 1. That is, here, we have, $a=b=0,p=1$ . Then the equation of the cylinder in cartesian coordinates is,

$x^(2)+y^(2)=1$

Now, the spherical coordinate system is represented by $(\rho,\theta,\phi)$

The relation between cartesian and spherical coordinates is given by,

$x=\rho sin\phi cos\theta,y=\rho sin\phi sin\theta, z= \rho cos\phi$

Thus, the equation of the cylinder can be rewritten in spherical coordinates as,

$(\rho sin\phi cos\theta)^(2)+(\rho sin\phi sin\theta)^(2)=1$

$\Rightarrow \rho^(2) sin^(2)\phi cos^(2)\theta+\rho^(2) sin^(2)\phi sin^(2)\theta=1$

$\Rightarrow \rho^(2) sin^(2)\phi (cos^(2)\theta+sin^(2)\theta)=1$

$\Rightarrow \rho^(2) sin^(2)\phi=1$ (As $sin^(2)\theta+cos^(2)\theta=1$ )

Note that $\rho$ represents the distance of a point from the origin, which is always positive. $\phi$ represents the angle made by the line segment joining the point with z-axis. The range of $\phi$ is given as $0\leq \phi\leq \pi$ . We know that in this range the sine function is positive. Thus, we can say that $sin\phi$ is always positive.

Thus, we can square root both sides and only consider the positive root as,

$\Rightarrow \rho sin\phi=1$

This is the required equation of the cylinder in spherical coordinates.

Final answer:

(a) The equation of the given sphere in cylindrical coordinates is $r^(2)+z^(2)=25$

(b) The equation of the given cylinder in spherical coordinates is $\rho sin\phi=1$

Mike buys a pack of markers for $2.85. He uses a 20% off coupon, what is the price of the markers with the coupon?A $2.28
B $3.42
C .57
D $2.53

Answers

The answer would be C

Answer:

c.57

Step-by-step explanation:

A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. The numbers y of cell sites from 1985 through 2011 can be modeled byy = 269573/1+985e^-0.308t where t represents the year, with t = 5 corresponding to 1985. Use the model to find the numbers of cell sites in the years 1998, 2003, and 2006.