Answer:
together they rake 
leaves in one minute.
Step-by-step explanation:
If Maya takes x minutes to rake, then in one minute, she rakes 
leaves.
For Carla, if her raking takes y minutes, in one minute she rakes 
leaves.
Thus, to find the amount they work together in one minute, we add both of their contributions for that minute, resulting in: 
Finally, simplifying this expression gives:
As a result, they collectively rake leaves in a minute.
The question is:
Examine a differential equation expressed as
y′ = f(αt + βy + γ),
where α, β, and γ are constants. Employ the variable change
z = αt + βy + γ to reformulate the differential equation as a separable equation of the type z′ = g(z).
Answer:
The equation
y′ = f(αt + βy + γ)
can be rephrased as
dy/dt = f(αt + βy + γ).
Our goal is to rewrite this differential equation in the form
z' = g(z), that is dz/dt = g(z).
First, be aware that
dz/dt = (dz/dy) * (dy/dt)...................(1)
Utilizing the substitution
z = αt + βy + γ
as specified,
dz/dy = β..........................................(2)
dy/dt = f(αt + βy + γ) = f(z)............(3)
From equations (2) and (3),
dz/dt = β.f(z) = g(z)
Thus,
z' = g(z)
Where g(z) = βf(z).
The formula for the volume of a sphere can be derived as follows. We will approach this through calculus, utilizing the concept of a solid of revolution; this is a three-dimensional shape formed by rotating a two-dimensional curve around a straight line (the axis of revolution) that lies within the same plane. From calculus, we know that we will generate a shape by rotating the specified circumference. Next, we isolate y and utilize certain limits for this integral.
