Chapter 6: Q. 92 (page 412)

Interstate_Highway_standards and Ohio Revised Code

Short Answer

Expert verified

(a) The required sin function $y = 15 \sin (\frac{π}{28} t)$

(b) The height of the tunnel at the edge of road is $\frac{15}{\sqrt{2}}$ units

Step by step solution

Step 1.(a) Find an equation for the sine curve that fits the opening. Place the origin at the left end of the sine curve.

The equation for the sine curve that fits the opening can be found out if we consider the curve with left end as the origin and half cycle as the half curve of sine function above the x-axis as follows-:

Now, the half-curve begins at $x = 0$ and ends at $x = 28$ , so the period of the function will be $T = 56$ as the full cycle will end at $x = 56$ . Therefore,

$ω = \frac{2 π}{T} = \frac{2 π}{56} = \frac{π}{28}$

Also, the amplitude of the function will be

A = 15

because the maximum value of this sine function at

x = 0

is 15 units.

Since the sine function is of the form

y = A \sin (ω t)

, we get the required sine function as-:

localid="1646480248236" $y = 15 \sin (\frac{π}{28} t)$

Step 2.(b) If the road is 14 feet wide with 7 -foot shoulders on each side, what is the height of the tunnel at the edge of the road?

If the road is 14 feet wide, the road will be 7 feet away from the origin point from where the sine curve starts.

So, we need to find the value of function at

x = 7

to find the height of tunnel at the edge of road which is-:

$y = 15 \sin (\frac{π}{28} (7)) = 15 \sin (\frac{π}{4}) = 15 (\frac{1}{\sqrt{2}}) = \frac{15}{\sqrt{2}}$

Hence,the height of the tunnel at the edge of road is $\frac{15}{\sqrt{2}}$ units

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Step 1.(a) Find an equation for the sine curve that fits the opening. Place the origin at the left end of the sine curve.

Step 2.(b) If the road is 14 feet wide with 7 -foot shoulders on each side, what is the height of the tunnel at the edge of the road?

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Decision Maths

Discrete Mathematics

Applied Mathematics

Study anywhere. Anytime. Across all devices.