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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Highway Design A section of highway connecting two hillsides with grades of 6% and 4% is to be built between two points that are separated by a horizontal distance of 2000feet (sec figure). At the point where the two hillsides come together, there is a 50-foot difference in elevation.

images

(a). Find the cubic function

f ( x ) = a x 3 + b x 2 + c x + d , 1000 x 1000

that describes the section of highway connecting the hillsides. At points A and B, the slope of the model must match the grade of the hillside.

(b) Use a graphing utility to graph the model.

(c) Use a graphing utility to graph the derivative of the model.

(d) Determine the grade at the steepest part of the transitional section of the highway.

(a)

To determine

To calculate: The cubic function that describes the highway between the hillsides where:

images

Explanation

Given:

The figure:

Calculation:

The provided information can be illustrated as:

Now, the equation of line OA would be:

y=601000x=0.06x

Thus, the slope of the line is 0.06.

The equation of the line BC would be:

y50=905010000(x0)y=0.04x+50

Thus, the slope of the line is 0.04.

Consider the following cubic polynomial:

f(x)=ax3+bx2+cx+d

It is provided that the elevation is 60 when x is 1000 and 90 when x is 1000 and 50 when x is 0.

This implies:

f(1000)=a(1000)3+b(1000)2+c(1000)+d60=10003a+10002b1000c+df(1000)=a(1000)3+b(1000)2+c(1000)+d90=10003a+10002b+1000c+d

And,

f(0)=a(0)3+b(0)2+c(0)+d50=d

The above equations lead to the following simplified equations:

(b)

To determine

To graph: The polynomial obtained f(x)=1.25×108x3+0.000025x2+0.0275x+50.

(c)

To determine

To graph: The derivative of the polynomial:

f(x)=1.25×108x3+0.000025x2+0.0275x+50.

(d)

To determine
The grade at the part of the road that is steepest where:

images

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