Problem 53 - College Algebra

Question

An architect designs two houses that are shaped and positioned like a part of the branches of the hyperbola whose equation is \(625 y^{2}-400 x^{2}=250,000,\) where \(x\) and \(y\) are in yards. How far apart are the houses at their closest point?

Step-by-Step Solution

Answer

The houses are 50 yards apart at their closest point.

Step 1: Rewrite the Hyperbola's Equation

First, we need to rewrite the equation of the hyperbola to isolate \(y\). This is done by dividing the whole equation by \(250,000\) and moving the \(x\) term to the other side of the equation. Doing so yields \[(y^2) = \frac{400(x^2) + 250,000}{625}\] and so, \(y = \pm\sqrt{\frac{400x^{2} + 250,000}{625}}= \pm\sqrt{\frac{4x^{2} + 625}{25}}.\) Here, the positive square root represents the upper branch of the hyperbola and the negative square root represents the lower branch.

Step 2: Calculate the Closest Points

The houses are positioned at two closest points of the hyperbola branches, meaning they are directly above or below each other. We do not care about the \(x\) value, since the houses are at the same horizontal position. The \(y\) value at this position is found by substituting \(x=0\) into the isolated equation from step 1, giving \(y = \pm\sqrt{625}\).

Step 3: Determine the Distance

The distance between the houses is the difference between the y-coordinates of the two houses. Since the houses are on opposite branches of the hyperbola, the distance is given by \[ 2\sqrt{625} = 50. \]