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USPshnik [31]
3 years ago
14

) Set up a double integral for calculating the flux of F=3xi+yj+zk through the part of the surface z=−5x−2y+2 above the triangle

in the xy-plane with vertices (0,0), (0,2), and (2,0), oriented upward. Instructions: Please enter the integrand in the first answer box. Depending on the order of integration you choose, enter dx and dy in either order into the second and third answer boxes with only one dx or dy in each box. Then, enter the limits of integration and evaluate the integral to find the flux.
Mathematics
1 answer:
Fynjy0 [20]3 years ago
8 0

The surface (call it S) is a triangle with vertices at the points

x=0,y=0\implies z=2\implies(0,0,2)

x=0,y=2\implies z=-2\implies(0,2,-2)

x=2,y=0\implies z=-8\implies(2,0,-8)

Parameterize S by

\vec s(u,v)=(1-v)(2,0,-8)+v\bigg((1-u)(0,2,-2)+u(0,0,2)\bigg)=(2-2v,2v-2uv,-8+6v+4uv)

with 0\le u\le1 and 0\le v\le1. Take the normal vector to S to be

\vec s_v\times\vec s_u=(20v,8v,4v)

Then the flux of \vec F across S is

\displaystyle\iint_S\vec F(x,y,z)\cdot\mathrm d\vec S=\int_0^1\int_0^1\vec F(x(u,v),y(u,v),z(u,v))\cdot(\vec s_v\times\vec s_u)\,\mathrm du\,\mathrm dv

=\displaystyle\int_0^1\int_0^1(6-6v,2v-2uv,-8+6v+4uv)\cdot(20v,8v,4v)\,\mathrm du\,\mathrm dv

=\displaystyle8\int_0^1\int_0^1(11v-10v^2)\,\mathrm du\,\mathrm dv=\boxed{\frac{52}3}

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marusya05 [52]
X would equal 70°.

For every triangle, all interior angles add up to 180°. This means that we can add together 42 and 68 to get 110°. Because the sum of all interior angles in a triangle must be 180°, we subtract 110 from 180 to see how much more we need, which gets us 70°.

I hope this helps!
6 0
3 years ago
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A box holding pennies, nickels and dimes contains thirteen coins with a total value of 83 cents. How many coins of each type are
Art [367]

Answer:

The number of pennies,nickels and dimes are (p,n,d)=(3,4,6).

Further explanation:

Given:

A box holding pennies, nickels and dimes contains thirteen coins in a box.

Total value is 83 cents.

Calculation:

Consider p,n and d be the number of pennies, nickel and dimes.

Now, total is 13 coins so p+n+d=13

As we know that these following are US coin.

Penny=1 cent

Nickel=5 cents

Dime=10 cents

Step 1:

The value is already given as 83 cents that is 80+3 cents.

80 cents can be possible in many combinations as follows:

(N,D)=(0,8),(2,7),(4,6),(6,5),(8,4),(10,3),(12,2),(14,1),(16,0)

It is given that the total number of cents is 13 so we choose (4,6) as (n,d) .

So the value of nickel n=4

Dimes d=6

Step 2:

The value of p is calculated as follows:

Substitute 4 for n, 6 for d in equation p+n+d=13 as follows:

p+4+6=13

p+10=13

p=13-10

p=3

Thus, the number of pennies,nickels and dimes are (p,n,d)=(3,4,6).

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For the sequence 3, 8, 13, 18, 23, ….: (i) Find recursive and non-recursive formulas. (ii) Prove the non-recursive formula using
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Answer:

Step-by-step explanation:

Given is the sequence 3, 8, 13, 18, 23, ….:

We find that this is a geometric sequence with each term added with 5 to the previous term

Hence recursive formula is a_n =a_{n-1} +5

Non recursive formula:

a_n =a_{n-1} +5=a_{n-2} +2(5)\\=a_{n-3} +3(5)-...\\=a_{n-(n-1)} +5(n-1)\\=a_1+(n-1)5\\=3+5n-5\\=5n-2

iii) Start with 3.

Add 5 and write as 2nd term

Take the resulting term and add 5 and mark as 3rd term

Repeat this m times.

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