Answer:
Step-by-step explanation:
Volume = 
find partial derivatives using product rule
i.e.
Using maximum for partial derivatives, we equate first partial derivative to 0.
y=0 or x+y =6
x=0 or x+4y =12
Simplify to get y =2, x = 4
thus critical points are (4,2) (6,0) (0,3)
Of these D the II derivative test gives
D<0 only for (4,2)
Hence maximum volume is when x=4, y=2, z= 4/3
Max volume is = 4(2)(4/3) = 32/3
With amounts measured in gallons, let
x = amount of 65% antifreeze
y = amount of 90% antifreeze
1 gal of the 65% brand contains 0.65 gal of pure antifreeze; x gal would contain 0.65x gal. Similarly, y gal of the 90% brand contains 0.90y gal of pure antifreeze.
To obtain 120 gal of 80% antifreeze solution (which contains 0.80•120 = 96 gal of pure antifreeze), we must have
x + y = 120 … … … … … [total volume of antifreeze solution]
0.65x + 0.90y = 96 … [total volume of pure antifreeze]
Solve the first equation for y :
y = 120 - x
Substitute this into the second equation and solve for x :
0.65x + 0.90 (120 - x) = 96
0.65x + 108 - 0.90x = 96
0.25x = 12
x = 48
Solve for y :
y = 120 - 48
y = 72
Hello there! So, the item has a 20% discount, but Juan still had to pay 80% for the item. To find the amount he paid, all you have to do is multiply the price by the percentage. We multiply by 80%, because he still had to pay the portion of the price. 80 * 80% (0.8) is 64. The. Juan paid $64.
Let
. Then
and
are two fundamental, linearly independent solution that satisfy
Note that
, so that
. Adding
doesn't change this, since
.
So if we suppose
then substituting
would give
To make sure everything cancels out, multiply the second degree term by
, so that
Then if
, we get
as desired. So one possible ODE would be
(See "Euler-Cauchy equation" for more info)
Answer:
Check attachment.
Step-by-step explanation:
I graphed it.
