Help me pls with this
1 answer:
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When solving these proportions we just remember when moving a number from one side to the other if it started in the numerator it ends up in the denominator and vice versa.
I'll do it in two steps here for teaching purposes; it's not too hard to go directly to the answer.
Answer:
The volume of the largest rectangular box (V) = 81/4
Step-by-step explanation:
<u>Step 1</u>:-
Given volume of the largest rectangular box in the first octant
V = l b h
let (x ,y, z) be the one vertex in the given plane
V = f(x, y, z) = x y z
given plane. Ф (x, y, z) = x + 9y + 4z = 27 ........(1)
<u>Step( ii):</u>
By using Lagrange multipliers
Suppose it is required to find the extreme for the function f(x, y, z) subject to the condition Ф (x, y, z) =0
Form Lagrange function F(x, y, z) = f(x, y, z) + λ Ф (x, y, z) where λ is called the Lagrange multipliers
F(x, y, z) = x y z+ λ ( x + 9y + 4z - 27) ......(2)
Obtain the equations are δ F / δ x = 0
δ f / δ x +λ δ Ф / δ x =0
⇒ y z + λ ( 1) = 0
⇒ y z = - λ ......(a)
Obtain the equations are δ F / δ y = 0
δ f / δ x +λ δ Ф / δ x =0
⇒ x z + λ ( 9) = 0
⇒ = - λ .......(b)
Obtain the equations are δ F / δ z = 0
δ f / δ x +λ δ Ф / δ z =0
x y + λ ( 4) = 0
⇒ = - λ .......(c)
<u>Step (iii):-</u>
Equating (a) and (b) equations
we get
cancel 'z' value we get x = 9y .......(d)
Equating (b) and (c) equations
we get
cancel 'x' value on both sides , we get
4z =9y ......(e)
substitute (d) (e) values in equation (1) we get
x + 9y + 4z -27 = 0
9y + 9y + 9y -27 =0
27y -27 =0
<u> y = 1</u>
substitute y =1 in x = 9y
<u>x = 9</u>
substitute y =1 in 4z =9y
<u>z = 9 /4</u>
therefore the dimensions are x =9 , y=1 and z = 9 /4
<u>Conclusion</u>:-
The largest volume of the rectangular box V = x y z
substitute x =9 , y=1 and z = 9 /4
V = 9(1)(9/4) =81/4
The largest volume of the rectangular box (V) = 81/4
Verification :-
Given plane x + 9y + 4z = 27
substitute x =9 , y=1 and z = 9 /4
9 + 9 + 4(9/4) = 27
27 =27
so satisfied equation