4x + y =3 x + y - z = 9 2x + 2y + z = 0
1 answer:
Answer:
Step-by-step explanation:
{4x + y = 3
{x + y - z = 9 ←
{2x + 2y + z = 0 ←
{4x + y = 3
{3x + 3y = 9 >> Combined equation
−¾[4x + y = 3]
{−3x - ¾y = −2¼ >> New Equation
{3x + 3y = 9
_________
2¼y = 6¾
___ ___
2¼ 2¼
[Plug this back into all three equations to get the z-value of −6, and the x-value of 0];
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Answer:
The volume is
Step-by-step explanation:
* Lets talk about the shell method
- The shell method is to finding the volume by decomposing
a solid of revolution into cylindrical shells
- Consider a region in the plane that is divided into thin vertical
rectangle
- If each vertical rectangle is revolved about the y-axis, we
obtain a cylindrical shell, with the top and bottom removed.
- The resulting volume of the cylindrical shell is the surface area
of the cylinder times the thickness of the cylinder
- The formula for the volume will be:
where 2πx · f(x) is the surface area of the cylinder shell and dx is its
thickness
* Lets solve the problem
- To find the volume V generated by rotating the region bounded
by the curves y = 4e^x and y = 4e^-x about the y-axis by use
cylindrical shells
- Consider that the height of the cylinder is y = (4e^x - 4e^-x)
- Consider that the radius of the cylinder is x
- The limits are x = 0 and x = 1
- Lets take 2π and 4 as a common factor out the integration
∴
∴
- To integrate and
we will use
integration by parts methods
∵ u = x
∴ u' = du/dx = 1 ⇒ differentiation x with respect to x is 1
∵ v' = dv/dx = e^x
- The integration e^x is e^x ÷ differentiation of x (1)
∴
∴
- Similar we will integrate xe^-x
∵ u = x
∴ u' = du/dx = 1
∵ v' = dv/dx = e^-x
- The integration e^-x is e^x ÷ differentiation of -x (-1)
∴
∴
∴ V = from 0 to 1
- Lets substitute x = 1 minus x = 0
∴
∴
∵
∴
Let t be time and r be rate, then if time varies inversely with the rate, the equation is 
. If it takes 5 hours to drive a fixed distance at a rate of 80, we can sub those values in to solve for the constant of variation, k.
. Solve for k by multiplying 5 and 80 to get that k = 400. Now let's find a new time t when r is a rate of 70. We will use that k value to do this: 
and find that it will take 20 hours to drive the distance at 70 mph when it takes 5 hours to drive the distance at 80 mph. Makes sense that it takes longer to drive a fixed distance when you are going slower.