Question

A railroad tunnel is shaped like a semi-eclipse. The height of the tunnel is 28ft and the total width is 50ft. Find the vertical clearance at a point 13ft to the right from the center. Please show all work that leads to your answer. Round your final answer to the nearest tenth.

Answers

Let the ellipse be modeled by

x^2/a^2 + y^2/b^2 = 1

Since the base has width 50, and the height is 28, we have

b = 28
a = 25

and the ellipse is

x^2/25^2 + y^2/28^2 = 1

When x=13, y=23.9
Standard form equation of an ellipse:

x² / a² + y² / b² = 1

where:

a is the semimajor axis

b is the semiminor axis


In this case:

a = total width / 2 = 50ft / 2 = 25ft

b = The height = 28ft


x² / a² + y² / b² = 1

x² / 25² + y² / 28² = 1


The vertical clearance at a point 13ft mean:

x = 13


x² / 25² + y² / 28² = 1

13² / 25² + y² / 28² = 1

Subtract 13² / 25² to both sides

13² / 25² + y² / 28² *- 13² / 25² = 1 - 13² / 25²

y² / 28² = 1 - 13² / 25²

y² / 28² = 25² / 25² - 13² / 25²

y² / 28² = ( 25² - 13² ) / 25²

Multiply both sides by 28²

y² = 28² ∙ ( 25² - 13² ) / 25²

Take the square root of both sides

y = ± √ [ 28² ∙ ( 25² - 13² ) / 25² ]

y = ± √ 28² ∙ √ ( 25² - 13² ) / √ 25²

y = ± 28 ∙ √ ( 25² - 13² ) / 25

y = ± 28 ∙ √ ( 625 - 169 ) / 25

y = ± 28 ∙ √ 456 / 25

y = ± 28 ∙ √ ( 4 ∙ 114 ) / 25

y = ± 28 ∙ √ 4 ∙ √ 114 / 25

y = ± 28 ∙ 2 ∙ √ 114 / 25

y = ± 56 ∙ √ 114 / 25

y = ± 56 ∙ 10.677078252 / 25

y = ± 597.916382112 / 25

y = ± 23.91665528448


y = ± 23.9 ft

rounded to the nearest tenth

Related Questions