Variable Cross Section Container

Calculus Level 3

A container has a variable elliptical cross section. These cross sections are parallel to the $xy$ plane, and the height of the container is along the $z$ axis.

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \\ a(z) = 1 + \alpha z \\ b = 1 \\ 0 \leq z \leq 5$

The bottom of the container has a circular base, and is located at $z = 0$ . The top of the container is located at $z = 5$ . The semi-minor axis length $b$ is constant, and the semi-major axis length $a$ is a function of $z$ . The positive constant $\alpha$ describes the rate of increase in the semi-major axis length.

If the volume of the container is $20$ , what is the value of $\alpha$ ?

This problem was inspired by the bottle for one of my favorite brands of "apple juice".

The answer is 0.1093.

1 solution

Karan Chatrath
Jan 13, 2021

The area of an ellipse of semi axes $a$ and $b$ is $A = \pi ab$ . At a general height $z$ , the elementary volume of an elliptic slab of thickness $dz$ can be written as:

$\blue{dV = A \ dz} \implies \blue{dV = \pi ab \ dz} \implies \blue{dV = \pi(1 + \alpha z) \ dz}$ $\blue{V =20= \int_{0}^{5} \pi(1 + \alpha z) \ dz}$ $\implies \boxed{\alpha = \frac{8}{5 \pi} - \frac{2}{5}}$

Variable Cross Section Container

The answer is 0.1093.

1 solution

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