Find how fast (a) the circumference (b)the area, if a cicle increases when the radius increases.
For absolute change rates:
dC/dt = 2*pi*dR/dt
dA/dt = 2*pi*R*dR/dt
For relative change rates
(1/C)dC/dt = (1/R) dR/dt
(1/A) dA/dt = (2/R)*dR/dt
Oh, the speed of circles! Well, (a) if the radius of a circle increases, the circumference increases too. It's like when you give a circle a little stretch, its waistline expands! And (b) if the radius increases, the area also increases. It's like the circle is making room for more pizza toppings, yum! So, the bigger the radius, the zestier the circumference and the larger the area. Keep those circles expanding!
To find out how fast the circumference and area of a circle increase when the radius increases, we can use calculus. Let's start with the circumference:
(a) The formula for the circumference of a circle is C = 2πr, where C represents the circumference and r is the radius.
To determine how the circumference changes with respect to the radius, we take the derivative of the circumference equation with respect to r:
dC/dr = d(2πr)/dr = 2π
This means that the derivative of the circumference with respect to the radius is a constant, 2π. So, regardless of the value of the radius, the circumference of a circle increases at a constant rate of 2π units for every unit increase in the radius.
(b) Now let's find out how the area of a circle changes when the radius increases.
The formula for the area of a circle is A = πr^2, where A represents the area and r is the radius.
To determine how the area changes with respect to the radius, we take the derivative of the area equation with respect to r:
dA/dr = d(πr^2)/dr = 2πr
This means that the derivative of the area with respect to the radius is 2πr. So, the rate at which the area increases depends on the radius itself. As the radius increases, the rate of increase in the area becomes larger.
In summary:
(a) The circumference of a circle increases at a constant rate of 2π units for every unit increase in the radius.
(b) The rate at which the area of a circle increases depends on the radius itself, following the equation dA/dr = 2πr.
To find how fast the circumference and area of a circle increase when the radius increases, we can use derivatives. The derivative of a function gives us the rate of change of that function with respect to a specific variable.
(a) To find how fast the circumference (C) increases with respect to the radius (r), we can differentiate the circumference formula with respect to the radius:
C = 2πr
Differentiating both sides of the equation with respect to r:
dC/dr = d(2πr)/dr
The derivative of 2πr with respect to r is simply 2π, as the derivative of r is 1.
So, dC/dr = 2π
This means that the circumference increases at a constant rate of 2π for every unit increase in the radius.
(b) To find how fast the area (A) increases with respect to the radius (r), we can differentiate the area formula with respect to the radius:
A = πr^2
Differentiating both sides of the equation with respect to r:
dA/dr = d(πr^2)/dr
Using the power rule of differentiation, we bring down the exponent of r and multiply it with the coefficient π:
dA/dr = 2πr
This means that the area increases with respect to the radius at a rate of 2πr. The rate of change of the area is linearly proportional to the radius.
So, to summarize:
(a) The circumference increases at a constant rate of 2π for every unit increase in the radius.
(b) The area increases at a rate of 2πr with respect to the radius, where r is the current radius value.