# Volumes - Linear Relationships

• Jul 17th 2010, 08:34 PM
UltraGirl
Volumes - Linear Relationships
Hi

For this question:

A symmetrical pier of height 5 m has an elliptical base with equation x^2/25 + y^2/4 = 1 and slopes to a parallel elliptical top with equation x^2/9 + y^2 = 1. If the cross sections of the area parallel to the base are also elliptical find the volume of the pier given that the area of an ellipse with semi-major axis a and semi-minor axis b is pi * a *b.

How exactly do I find a and b in terms of h? I know I need to do something with linear relationships but I've forgotten the method. Is it something to do with y = ax+b?
• Jul 17th 2010, 08:50 PM
Failure
Quote:

Originally Posted by UltraGirl
Hi

For this question:

A symmetrical pier of height 5 m has an elliptical base with equation x^2/25 + y^2/4 = 1 and slopes to a parallel elliptical top with equation x^2/9 + y^2 = 1. If the cross sections of the area parallel to the base are also elliptical find the volume of the pier given that the area of an ellipse with semi-major axis a and semi-minor axis b is pi * a *b.

How exactly do I find a and b in terms of h? I know I need to do something with linear relationships but I've forgotten the method. Is it something to do with y = ax+b?

Let $\displaystyle a(z)$ be the major half-axis, $\displaystyle b(z)$ the minor half-axis at hight $\displaystyle z$.
You know that $\displaystyle a(0)=5, b(0)=2$ and $\displaystyle a(h)=3, b(h)=1$. Assuming that $\displaystyle a(z), b(z)$ are linear functions of $\displaystyle z$, you get that
$\displaystyle a(z)=a(0)+\frac{a(h)-a(0)}{h}\cdot z$ and $\displaystyle b(z)=b(0)+\frac{b(h)-b(0)}{h}\cdot z$

Now integrate the cross section $\displaystyle A(z) := \pi a(z)\cdot b(z)$ from $\displaystyle z=0$ to $\displaystyle z=h$.