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2 years ago

What is the Greatest Common Factor and the Least common Multiple os 3 and 8

Mathematics

2 answers:

Tatiana [17]2 years ago

7 0

3 is a prime number - the GCF is 1, and the least common multiple of 3 and 8 is 24.

Katen [24]2 years ago

3 0

Answer:

Greatest Common Factor is 1

Least Common Multiple is 24

Step-by-step explanation:

Greatest common factor is the biggest whole number that can be multiplied into the numbers. In this case 1x8=8 and 1x3=3.

Least common multiple is the smallest number each number can multiply into. In this case 3x8=24 and 8x3=24

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How to find each angle indicated

Natalija [7]

The answer is 80 because all triangles equal 180

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2 years ago

Read 2 more answers

Name/ Uid:1. In this problem, try to write the equations of the given surface in the specified coordinates.(a) Write an equation

Gemiola [76]

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,

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

$r^{2}=x^{2}+y^{2},tan\theta=\frac{y}{x},z=z$

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 $\rho sin\phi=1$

7 0

3 years ago

The sum of the measures of the two angles of a linear pair must be...?

Rina8888 [55]

The answer is 180 degrees

5 0

2 years ago

Explain in easy terms, how the signs of the last terms of the two binomial factors of a trinomial are determined?

shutvik [7]

Explanation:

It helps to understand the process of multiplying the binomials. Consider the simple case ...

(x +a)(x +b)

The product is ...

(x +a)(x +b) = x² +(a+b)x + ab

If the constant term (ab) is negative, the signs of (a) and (b) are different.

If the constant term (ab) is positive, the signs of (a) and (b) will both match the sign of the coefficient of the linear term (a+b).

___

Of course, the sum (a+b) will have the sign of the (a) or (b) value with the largest magnitude, so when the signs of (a) and (b) are different, the factor with the largest magnitude will have the sign of (a+b), the x-coefficient.

Example:

x² -x -6

-6 tells you the factors will have different signs. -x tells you the one with the largest magnitude will be negative.

-6 = -6×1 = -3×2 = ... (other factor pairs have a negative factor with a smaller magnitude)

The sums of these factor pairs are -5 and -1. We want the factor pair that has a sum of -1, the coefficient of x in the trinomial.

x² -x -6 = (x -3)(x +2)

5 0

3 years ago

How are the numbers 3, 4, and 17 alike

Tanzania [10]

They are all prime numbers

4 0

2 years ago