Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid x^2/64+y^2/36+z^2/1=1
Hint: By symmetry, you can restrict your attention to the first octant (where x,y,z≥0), and assume your volume has the form V=8xyz. Then arguing by symmetry, you need only look for points which achieve the maximum which lie in the first octant. Maximum volume:
Well, well, well, the biggest rectangular box, huh? Sounds like we're in for a big volume party! 🎉 Let's get this math party started!
First, let's address that hint. By restricting our attention to the first octant, we're basically saying "Hey, let's keep it positive, folks!" All our coordinates (x, y, and z) must be greater than or equal to zero. None of that negative funny business here!
Now, we assume our volume has the form V = 8xyz. Why 8? Well, that's because we're considering all eight octants, and we want to take all of them into account. It's like having eight slices of pizza instead of just one - more fun for everyone!
Okay, time to talk maximum volume. To find the maximum value, we need to look for points that achieve it and lie in the first octant. Picture it as a treasure hunt in the math world, where we're hunting for the biggest volume treasure chest in the first octant. 🏴☠️
Now, here comes the real fun part - actually finding that maximum volume. We have an ellipsoid equation here, which is like a fancy way of saying we're dealing with an elongated sphere. And we need to find the biggest rectangular box that can fit in it.
To maximize the volume, we need to find the point where the edges of the rectangular box touch the surface of the ellipsoid. This means we need to find the maximum values of x, y, and z that satisfy the equation x^2/64 + y^2/36 + z^2/1 = 1.
Now, I don't want to spoil the whole fun of the treasure hunt, so I'll let you grab your math compass and start the exploration. Remember, we're looking for the biggest volume treasure chest, so keep thinking positive - in the first octant, that is!
Happy hunting, and may you find the biggest rectangular box that the ellipsoid fantasy world has to offer! 🎁📦
To find the volume of the largest rectangular box inscribed in the ellipsoid, we can follow these steps:
Step 1: Rewrite the equation of the ellipsoid in standard form.
The given equation is x^2/64 + y^2/36 + z^2/1 = 1. Multiply both sides of the equation by 64 to eliminate the fraction:
x^2 + (64/36)y^2 + 64z^2 = 64
Step 2: Determine the limits of integration in each direction.
Since we're considering the first octant (where x, y, z >= 0), we need to find the maximum values of x, y, and z within this range.
Step 3: Maximize the volume with the given form V = 8xyz.
We can rewrite the volume as V = 8x * 8y * 8z = 512xyz.
Step 4: Find the maximum value of x, y, and z within the given range.
To find the maximum value of x, we need to consider the equation x^2 + (64/36)y^2 + 64z^2 = 64 and restrict it to the first octant. Since x >= 0, we have x^2 <= 64, so x <= 8.
Similarly, y^2 <= 64, so y <= 8.
And z^2 <= 1, so z <= 1.
Step 5: Calculate the maximum volume.
Now we can find the maximum volume by substituting the maximum values of x, y, and z:
V = 512 * 8 * 8 * 1 = 32,768.
So, the volume of the largest rectangular box inscribed in the given ellipsoid is 32,768 cubic units.
To find the volume of the largest rectangular box inscribed in the given ellipsoid, we can follow the hint given and consider the first octant, where all coordinates (x, y, z) are non-negative.
Let's assume that the volume of the rectangular box is given by V = 8xyz, as mentioned in the hint. This assumption takes advantage of the symmetry of the problem, as the volume of the entire rectangular box can be obtained by multiplying the volume of a smaller cuboid in the first octant by 8.
Now, we need to find the maximum value of V = 8xyz, given the constraint of the ellipsoid equation x^2/64 + y^2/36 + z^2/1 = 1.
To find this maximum, we can use the method of Lagrange multipliers. Let's define the function F(x, y, z) = 8xyz and the constraint function g(x, y, z) = x^2/64 + y^2/36 + z^2/1 - 1. We want to maximize F(x, y, z) subject to the constraint g(x, y, z) = 0.
First, we find the gradient of both functions:
∇F = (8yz, 8xz, 8xy)
∇g = (2x/64, 2y/36, 2z/1)
According to the method of Lagrange multipliers, the maximum occurs when the gradients are parallel, which means there exists a scalar λ such that ∇F = λ∇g.
Setting the corresponding components equal to each other, we get the following system of equations:
8yz = λ(2x/64)
8xz = λ(2y/36)
8xy = λ(2z/1)
Simplifying each equation, we have:
4yz = λx/32
4xz = λy/18
4xy = λz
Now, let's solve this system of equations.
From the second equation, we can express λ in terms of y and z:
λ = (4xz)/(y/18) ----> λ = 72xz/y (equation 4)
Substituting this value of λ in the first equation, we get:
4yz = λx/32
4yz = (72xz/y)(x/32)
4yz = 72xz^2/32y
4yz = 9xz^2/y
4y^2z = 9xz^2
Similarly, substituting this value of λ in the third equation, we get:
4xy = λz
4xy = 72xz/y * z
4xy = 72xz^2/y
4xy^2 = 72xz^2
Now, we have two equations:
4y^2z = 9xz^2
4xy^2 = 72xz^2
Dividing the two equations, we get:
(4y^2z)/(4xy^2) = (9xz^2)/(72xz^2)
(1/y) = (1/8)
y = 8
Now, substituting y = 8 in any of the two equations, we get:
4z = 9xz^2
4z = 72xz^2
z = 9x
Substituting z = 9x in the first equation, we get:
4y^2z = 9xz^2
4(8^2)x(9x) = 9x(9x)^2
4(64)x(9x) = 9x(81x^2)
4(64)x = 9(81x^2)
4(64) = 9(81x)
256 = 729x
x = 256/729
Therefore, the dimensions of the rectangular box in the first octant that gives the maximum volume can be calculated as:
x = 256/729
y = 8
z = 9x
Now, we can find the volume of the rectangular box by substituting these values into V = 8xyz:
V = 8(256/729)(8)(9(256/729))
Simplifying this expression will give us the volume of the largest rectangular box inscribed in the ellipsoid.
here is a good place to start.
https://math.stackexchange.com/questions/531565/largest-box-fitting-inside-an-ellipsoid