Answer to Question #171143 in Analytic Geometry for Jon jay mendoza

Question

Read and analyze the problem below.

Solve and show your complete solution.

A semielliptical arch over a tunnel for a road through a mountain has a major axis of 100

feet, and its height at the center is 30 feet

(see figure) https://ibb.co/vLSmtVH

A) Find the standard form of the equation of the ellipse that represents the given problem.

B) Determine the height of the arch 5 feet from the edge of the tunnel.

Answer 1

a) length of major axis (2a) = 100 feet

length of minor axis (2b) = 60 feet

a = 50 feet

b = 30 feet

from,

$\dfrac{x²}{a²} + \dfrac{y²}{b²} = 1$

$\dfrac{x²}{50²} + \dfrac{y²}{30²} = 1$

900x² + 2500y² = 2250000

b) when the arch is 5 feet, x = 50 - 5

900(45)² + 2500y² = 2250000

900(2025) + 2500y² = 2250000

2500y² = 2250000 - 1822500

2500y² = 427500

y² = 171

y = 13.1 feet