The GCF of 2 and another number is 1. Describe the other number.

Find the mass of the solid paraboloid Dequals={(r,thetaθ,z): 0less than or equals≤zless than or equals≤8181minus−r2, 0less th

Lubov Fominskaja [6]

Answer:

M = 5742π

Step-by-step explanation:

Given:-

- Find the mass of a solid with the density ( ρ ):

ρ ( r, θ , z ) = 1 + z / 81

- The solid is bounded by the planes:

0 ≤ z ≤ 81 - r^2

0 ≤ r ≤ 9

Find:-

Find the mass of the solid paraboloid

Solution:-

- The mass (M) of any solid body is given by the following triple integral formulation:

$M = \int \int \int {p ( r ,theta, z)} \, dV\\\\$

- We can write the above expression in cylindrical coordinates:

$M = \int\limits\int\limits_r\int\limits_z {r*p(r,theta,z)} \, dz.dr.dtheta \\\\M = \int\limits\int\limits_r\int\limits_z {r*[ 1 + \frac{z}{81}] } \, dz.dr.dtheta\\\\$

- Perform integration:

$M = \int\limits\int\limits_r{r*[ z + \frac{z^2}{162}] } \,|_0^8^1^-^r^2 dr.dtheta\\\\M = \int\limits\int\limits_r{r*[ 81-r^2 + \frac{(81-r^2)^2}{162}] } \, dr.dtheta\\\\M = \int\limits\int\limits_r{r*[ 81-r^2 + \frac{6561 -162r + r^2}{162}] } \, dr.dtheta\\\\M = \int\limits\int\limits_r{r*[ 81-r^2 + 40.5 -r +\frac{r^2}{162} ] } \, dr.dtheta\\\\M = \int\limits\int\limits_r{[ 121.5r-r^2 -\frac{161r^3}{162} ] } \, dr.dtheta\\\\$

$M = 2*\int\limits_0^\pi {[ 121.5r^2-r^3 -\frac{161r^4}{162} ] } |_0^6 \, dtheta\\\\M = 2*\int\limits_0^\pi {[ 121.5(6)^2-(6)^3 -\frac{161(6)^4}{162} ] } \, dtheta\\\\M = 2*\int\limits_0^\pi {[ 4375-216 -1288] } \, dtheta\\\\M = 2*\int\limits_0^\pi {[ 2871] } \, dtheta\\\\M = 5742\pi kg$

- The mass evaluated is M = 5742π

8 0

2 years ago

the Following table shows alexandra's investment options over the course of three years. Her initial investment was $1,000 . wri

Y_Kistochka [10]

The pattern in the given series of amount in the account are in the form of

arithmetic and geometric progression.

The function for Option 1 is; $\underline{ f(n) = 1,100 + (n - 1) \cdot 100}$

The function for Option 2 is; $\underline{f(n) = 1,100 \times 1.1^{(n - 1)}}$

Reasons:

The given table of values is presented as follows;

$\begin{tabular}{c|c|c|c|}Number of years&1&2&3\\Option 1 (Amount in dollars)&1,100&1,200&1,300\\Option 2 (Amount in dollars)&1,100&1,210&1,331\end{array}\right]$

In Option 1, the amount in dollars for each year has a common difference of d = 100

The first term, a = 1,100

Therefore;

The Option 1 can be represented as an arithmetic progression , A.P. in the

form, tₙ = a + (n - 1)·d as follows;

For the Option 1, we have;

The amount in dollars after <em>n</em> years, $\underline{ f(n) = 1,100 + (n - 1) \cdot 100}$

For Option 2, it is possible to find;

1,331 ÷ 1,210 = 1,210 ÷ 1,100 = 1.1

Therefore;

The terms in the Option 2 have a common ratio of r = 1.1

The Option 2 is a geometric progression, G.P.

The first term in Option 2 is a = 1,100

Which gives, the nth term, tₙ = a·r⁽ⁿ ⁻ ¹⁾

Therefor;

The function for the Option 2 is; $\underline{f(n) = 1,100 \times 1.1^{(n - 1)}}$

Learn more about arithmetic and geometric progression here:

brainly.com/question/8932895

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4 0

2 years ago

Jane can type 150 words in 2 minutes. How many<br> words can she type in 5 minutes?

Masteriza [31]

I believe about 750 this is because of you multiply 150 and 5 you get your answer 750 because each 2 minutes she types is 150 and although you can find your average I just add the average 5 times this would be an easier way!!!

8 0

2 years ago

If 2 angles have the same angle measure they are said to be what

DanielleElmas [232]

2 angles that have the same angle measure are congruent.

Given that the two angles have the same angle measure.

Congruence is a term reserved for geometry. Two metrics are congruent when you can perfectly map to each other by mirroring, rotating, and translating without distortion.

We know that two shapes or objects are congruent if they have the same shape and the same size. For two angles to be congruent, they must coincide when they are overlapped. This is only possible if the two angles are equal.

Therefore, if two angles have the same measure, then they are said to be congruent.

Learn more about congruent angle from here brainly.com/question/2938476

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8 0

2 years ago

The first step in long division is to ___. divide multiply subtract

Aleonysh [2.5K]

Answer:

divide

Step-by-step explanation:

5 0

2 years ago

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