Mathematics

Problem

## 85. Derive an Algebraic Model of a Problem Show that the distance

x

in the figure is a solution of the equation

x^{4}-16 x^{3}+500 x^{2}-8000 x+32,000=0

and find the value of

D

by following these steps. ![](https://storage.googleapis.com/diraq-contents/question-bank/question-images/69cefc5635a6/2025_11_21_dcfe37ceeb51b74c3162g-12.jpg) (a) Use the similar triangles in the diagram and the properties of proportions learned in geometry to show that

\frac{8}{x}=\frac{y-8}{y} .

(b) Show that

y=\frac{8 x}{x-8}

. (c) Show that

y^{2}-x^{2}=500

. Then substitute for

y

, and simplify to obtain the desired degree 4 equation in

x

. (d) Find the distance

D

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