A closed cylindrical container is to be made for $8400. The material for the top

Question

A closed cylindrical container is to be made for $8400. The material for the top will cost $3 per in^2 and for bottom of the container will cost $4 per in^2, and the material for the sides will cost $7 per in^2. Find the dimensions of the container of the largest possible volume. The measurement of a side of a cube container is found to be 12 centimeters, with a possible error of 0.05 centimeter. Approximate the percent error in computing the volume of the container. Find the area of the region bounded by f(x) = (x + 1)(x - 2)^2 and x-axis.

Explanation / Answer

Solution:(6)

V = x^3
V / V = 3 x / x
= 3 (0.05) / 12
= 5/4 % 1.25 %

Solution:(7)

Since the x axis is where y = 0, then we have to find the roots of:
(x+1)(x-2)^2 = 0.
This is easily done by factoring;
x = -1, and 2

So if we do the integration

A = (x+1)(x-2)^2 dx from -1 to 2;

we get:

A = (x^3 - 3x^2 + 4) dx from -1 to 2;

   = {(x^4)/4 - x^3 + 4x} from -1 to 2;

   = {(2^4)/4 - (2)^3 + 4*2} - {((-1)^4)/4 - (-1)^3 + 4*(-1)}

   = {4 - 8 + 8} - {1/4 + 1 - 4}

   = 4 - 11/4

A = 5/4 or 1.25 unit^2

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