The a hollow cylinder. The internal and external radii are estimated to be 6 cm and 8 cm respectively, to the nearest whole number. The height of the cylinder is exactly 14 cm.
(a) Determine the exactly values for internal and external radii which give maximum volume of the materials used
(b) Calculate the maximum possible volume of the materials used. Take the value of π to be 22/7
a)2.5
To determine the exact values for the internal and external radii that give the maximum volume of the materials used, we can set up an equation and differentiate it with respect to the radii.
Let's denote the internal radius as r1 and the external radius as r2. The height of the cylinder, h, is given as 14 cm.
The volume of a hollow cylinder can be expressed as V = π * h * (r2^2 - r1^2).
(a) To find the exact values for the radii that give maximum volume, we can differentiate the volume equation with respect to the radii and set the derivatives equal to zero:
dV/dr1 = 0 and dV/dr2 = 0.
Differentiating the volume equation with respect to r1, we get:
dV/dr1 = π * h * (-2r1) = 0.
Solving this equation, we find that r1 = 0.
Differentiating the volume equation with respect to r2, we get:
dV/dr2 = π * h * (2r2) = 0.
Solving this equation, we find that r2 = 0.
However, these results are invalid since the radii cannot be zero. Therefore, there are no exact values for the internal and external radii that give the maximum volume of the materials used.
(b) We can still calculate the maximum possible volume of the materials used by using the estimated values for the radii.
Given that the estimated internal radius is 6 cm and the external radius is 8 cm, we can substitute these values into the volume equation:
V = π * h * (r2^2 - r1^2)
V = ((22/7) * 14 * (8^2 - 6^2)
V = (22/7) * 14 * (64 - 36)
V = (22/7) * 14 * 28
Calculating this expression, we find that the maximum possible volume of the materials used is approximately 12320 cm^3.