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professor190 [17]
3 years ago
14

In the following equation, a and b are both integers. a(3x-8)=b -18x

Mathematics
2 answers:
zaharov [31]3 years ago
8 0
The value of a is -6
The value of b is 48
I am Lyosha [343]3 years ago
6 0

If a and b are two fixed integers this means the equations is a straight line where x values could be any number within (-∞,∞). So we can evaluate any two numbers to obtain a set of two different equations to solve the system for a and b. The number we will try are 0 and 1:

For 0 (Equation #1)

a(3(0)-8)=b-18(0)\\a(-8)=b\\b=-8a

For 1 (Equation #2)

a(3(1)-8)=b-18(1)\\a(-5)=b-18\\-5a+18=b\\b=18-5a

We can solve the system by replacing equation #1 in equation #2 like this:

-8a=18-5a\\-8a+5a=18\\-3a=18\\a=\frac{18}{-3} \\a=-6

Now we can replace the value of a in equation #1

b=-8(-6)\\b=48

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Find the mass of the lamina that occupies the region D = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} with the density function ρ(x, y) = xye
Alona [7]

Answer:

The mass of the lamina is 1

Step-by-step explanation:

Let \rho(x,y) be a continuous density function of a lamina in the plane region D,then the mass of the lamina is given by:

m=\int\limits \int\limits_D \rho(x,y) \, dA.

From the question, the given density function is \rho (x,y)=xye^{x+y}.

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The mass of the lamina can be found by evaluating the double integral:

I=\int\limits^1_0\int\limits^1_0xye^{x+y}dydx.

Since D is a rectangular region, we can apply Fubini's Theorem to get:

I=\int\limits^1_0(\int\limits^1_0xye^{x+y}dy)dx.

Let the inner integral be: I_0=\int\limits^1_0xye^{x+y}dy, then

I=\int\limits^1_0(I_0)dx.

The inner integral is evaluated using integration by parts.

Let u=xy, the partial derivative of u wrt y is

\implies du=xdy

and

dv=\int\limits e^{x+y} dy, integrating wrt y, we obtain

v=\int\limits e^{x+y}

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I_0=xe^x

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I=\int\limits^1_0(xe^x)dx.

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I=\int\limits^1_0xe^xdx.

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I=\int\limits^1_0xe^xdx=0-1(0-1).

I=\int\limits^1_0xe^xdx=0-1(-1)=1.

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Therefore answer is 72 degrees.
4 0
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