Answer:
Volume of the Tetrahedron T =
Step-by-step explanation:
As given, The tetrahedron T is bounded by the planes x + 2y + z = 2, x = 2y, x = 0, and z = 0
We have,
z = 0 and x + 2y + z = 2
⇒ z = 2 - x - 2y
∴ The limits of z are :
0 ≤ z ≤ 2 - x - 2y
Now, in the xy- plane , the equations becomes
x + 2y = 2 , x = 2y , x = 0 ( As in xy- plane , z = 0)
Firstly , we find the intersection between the lines x = 2y and x + 2y = 2
∴ we get
2y + 2y = 2
⇒4y = 2
⇒y =
= 0.5
⇒x = 2(
) = 1
So, the intersection point is ( 1, 0.5)
As we have x = 0 and x = 1
∴ The limits of x are :
0 ≤ x ≤ 1
Also,
x = 2y
⇒y = 
and x + 2y = 2
⇒2y = 2 - x
⇒y = 1 - 
∴ The limits of y are :
≤ y ≤ 1 - 
So, we get
Volume = 
= ![\int\limits^1_0 {\int\limits^{1-\frac{x}{2}}_{y = \frac{x}{2}}{[z]}\limits^{2-x-2y}_0 {} \, \, dy \, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%5E1_0%20%7B%5Cint%5Climits%5E%7B1-%5Cfrac%7Bx%7D%7B2%7D%7D_%7By%20%3D%20%5Cfrac%7Bx%7D%7B2%7D%7D%7B%5Bz%5D%7D%5Climits%5E%7B2-x-2y%7D_0%20%7B%7D%20%5C%2C%20%20%20%5C%2C%20dy%20%20%5C%2C%20dx)
= 
= ![\int\limits^1_0 {[2y-xy-y^{2} ]}\limits^{1-\frac{x}{2}} _{\frac{x}{2} } {} \, \, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%5E1_0%20%7B%5B2y-xy-y%5E%7B2%7D%20%5D%7D%5Climits%5E%7B1-%5Cfrac%7Bx%7D%7B2%7D%7D%20_%7B%5Cfrac%7Bx%7D%7B2%7D%20%7D%20%7B%7D%20%5C%2C%20%5C%2C%20dx)
= ![\int\limits^1_0 {[2(1-\frac{x}{2} - \frac{x}{2}) -x(1-\frac{x}{2} - \frac{x}{2}) -(1-\frac{x}{2}) ^{2} + (\frac{x}{2} )^{2} ] {} \, \, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%5E1_0%20%7B%5B2%281-%5Cfrac%7Bx%7D%7B2%7D%20-%20%5Cfrac%7Bx%7D%7B2%7D%29%20%20-x%281-%5Cfrac%7Bx%7D%7B2%7D%20-%20%5Cfrac%7Bx%7D%7B2%7D%29%20-%281-%5Cfrac%7Bx%7D%7B2%7D%29%20%5E%7B2%7D%20%20%2B%20%28%5Cfrac%7Bx%7D%7B2%7D%20%29%5E%7B2%7D%20%5D%20%7B%7D%20%5C%2C%20%5C%2C%20dx)
= 
= 
= 1 - 1² +
- 0 + 0 - 0
= 1 - 1 +
= 
So, we get
Volume =
Answer:
y = 3x - 10
Step-by-step explanation:
Step 1: Rewrite 1st equation
2y = 6x + 4
y = 3x + 4
Step 2: Find 2nd equation
y = 3x + b
-4 = 3(2) + b
-4 = 6 + b
b = -10
Step 3: Rewrite 2nd equation
y = 3x - 10
And we have your parallel equation!
The simplification is 20x-15 because 2 times 6x is 12 x and plus the 8 x its 20 then 2 times -5 is -10 plus the other - 5 is -15
Answer:
8 minutes
Step-by-step explanation:
time = distance/speed
A decrease of 15 mph from 45 mph means the speed on the second segment is 30 mph. Filling in the given values in the above equation, we have ...
... time = (4 mi)/(30 mi/h) = 4/30 h = 8/60 h × (60 min/h) = 8 min
Answer:
9
Step-by-step explanation:
The answer to | − 12 + 3 | is 9 because if you add -12 + 3, you get -9. However, since it is the <em>absolute value</em>, we would make it the same number but positive.
Hope this helped! :)