# A straight road rises at an inclination of 0.3 radian from the horizontal

*label*Mathematics

*timer*Asked: Oct 16th, 2017

*account_balance_wallet*$10

**Question description**

Answer the following questions in a Word document and upload the document to the appropriate drop box.

1) A straight road rises at an inclination of 0.3 radian from the horizontal. Find the slope and change in elevation over a one-mile section of the road.

2) In order to allow rain to run off of a road, they are often designed with parabolic surfaces in mind. Suppose a road is 32 feet wide and 0.4 foot higher in the center than it is on the side

a) Write an equation of the parabola with its vertex at the origin

that models the road surface.

b) How far from the center of the road is the road surface 0.2 feet

lower than the center?

3. A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters. How wide is the archway at ground level?

4) Halley’s comet follows an elliptical orbit with the sun as one of the foci. The eccentricity of the orbit is approximately 0.967. The length of the major axis of the orbit is approximately 35.88 astronomical units. (An astronomical unit is about 93 million miles.

(a) Find an equation of the orbit. Let the origin represent the sun and

use the x axis as the major axis.

(b) Find the greatest and least distances (the aphelion and perihelion,

respectively) from the sun’s center to the comet’s center.

5) You and a friend live 4 miles apart. You hear the sound of thunder 18 seconds before your friend hears it. Assuming sound travels at 1100 feet

per second, determine where the lightning occur?

## Tutor Answer

Hello 😀I've prepared the solution for the assignment.I've included the complete process and concepts used for getting the answers.Please have a look at it and tell me if it requires any changes/improvements.Hoping for a positive response :)

Answer to Question 1

Figure for the problem

The slope of the road is given by,

Putting

,

From the figure, we can say that,

Putting,

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑟𝑜𝑎𝑑 = ℎ

,

Multiplying both sides by 1 mile,

Answers:

• Slope of the road is 0.309 (approx.)

• Change in elevation over a one-mile section of the road is

0.295 mile (approx.)

Answer to Question 2

Figure for the problem

The blue curve represents the parabolic road and the grey line represents the

ground level.

The general form of equation of a vertical parabola is,

Where (h,k) is the vertex of parabola.

From the figure we can see that the vertex is at (h,k)=(0,0).

Putting h=0 and k=0 in general form,

We’ve labeled the right extreme point of the road by A. The road is 16 feet wide

on each side of the centre. As the road is 0.4 feet lower at A than at the centre,

the coordinates of A will be (16,-0.4). As this point lies on the parabola, it should

satisfy its equation.

Putting x=16 and y=-0.4 in parabola’s equation,

Dividing both sides by -1.6,

Coming back to the ...

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