a cylindrical oil tank of radius r and volume V is closed at both and express the total surface area a in term of V and R if the volume is pi/4 * m ^ 3,
1)find the volume of r which makes the area a the minimum.
2)state the remainder theorem
1) To find the volume of r that makes the surface area a minimum, we need to differentiate the surface area formula with respect to r and set it equal to zero to find the critical points.
The formula for the surface area of a closed cylindrical tank is given by:
A = 2πr^2 + 2πrh
where r is the radius of the tank and h is the height.
Since the tank is closed, we can express the height in terms of the volume using the formula:
V = πr^2h
Solving for h, we get:
h = V / (πr^2)
Substituting this value of h into the surface area formula, we have:
A = 2πr^2 + 2πr(V / πr^2)
Simplifying, we get:
A = 2πr^2 + 2V / r
To find the minimum surface area, we need to differentiate A with respect to r and set it equal to zero:
dA/dr = 4πr - 2V / r^2 = 0
Solving for r:
4πr = 2V / r^2
2πr^3 = V
r^3 = V / (2π)
r = (V / (2π))^(1/3)
So, the volume of r that makes the surface area a minimum is given by r = (V / (2π))^(1/3).
2) The Remainder Theorem states that if a polynomial f(x) is divided by (x - c), then the remainder is equal to f(c).
In other words, if we have a polynomial f(x) and we divide it by (x - c), the remainder we get will be the value of f(c).
For example, if we have f(x) = 2x^3 - 3x^2 + 5x - 1 and we divide it by (x - 2), the remainder we get will be f(2).
The Remainder Theorem is essential in polynomial division and in finding roots of polynomials.