Wow, Lagrange multipliers in high school!
As a rule with these Lagrange multiplier problems, when the problem is symmetrical with respect to interchange of the variables, the solution almost always ends up with all the variables equal -- what else could it be?
We want to maximize the area of a rectangle with sides x and y subject to the perimeter being constant.
(i)
The area of a rectangle is just the product of its sides:
A = f(x,y) = xy
(ii)
The perimeter of a rectangle is the sum of its sides:
P = g(x,y) = x + x + y + y = 2x+2y
(iii)
Usually I like to form the objective function E=f-λg before I take the derivatives. I usually use a lambda not a gamma for the multiplier.
Let's do what they ask. They want the gradient ∇f(x, y)
∇f(x, y) = (y, x)
(iv)
λ∇g(x, y) = (2λ, 2λ)
(v)
I'm not sure what γ=1/2y is about; I'll solve it like I know how and see where we are.
There it is. We get
y = 2λ
so we also find
x = 2λ
(vi)
We have y=x=2λ so we've shown the variables are equal, i.e. our rectangle is a square. We can solve for λ using our constraint:
P = 2x+2y = 8λ
λ=P/8
so as expected we have a square with side length P/4:
x=y=2λ=P/4
Answer: the y intercept is 0
Step-by-step explanation:
Answer:
volume is 9847.04
Step-by-step explanation:
V = π * * h
V = 3.14 * * 16
=9847.04
A
First, I’m going to the line is question into slope intercept form.
y + 10 = -5x + 5
y = -5x + 5 -10
y = -5x -5
y + 10 = -5 (x - 1) becomes y = -5x -5 in slope intercept form. I will call this line ‘line 1’
A becomes y = 5x -15 in slope intercept form
B becomes y = -5x + 25 in slope intercept form
C becomes y = -5x -5 in slope intercept form
D becomes y = -5x + 10 in slope intercept form
C is the same line as line 1. Any point that is on line 1 is also on line C, so C cannot be it.
Notice that line B and D have the same slope but different y-intercepts as line 1. That means these lines are parallel (not the same line though - different y intercepts) to line 1, so they will never intercept.
Line A has a different slope vs line 1, so they will eventually intersect only once
4 minus the quotient of a number and 3