please explin A house is going to be built in a rural part of Oregon and the own

Question

please explin

A house is going to be built in a rural part of Oregon and the owners want the house to be away from the only road in the area. However, they do need a driveway and they want to minimize the costs of building it! An approximate formula that describes the nearby road is f(x) = 1/2x and the location of their house is at the point (0, 0.5). (See the figure below.) Find the coordinate on the road that is closest to the coordinate where the house is located. If the road is going to be a straight line connecting the two points, how long is the road? Write an equation that will describe the path of the driveway.

Explanation / Answer

a.)

closest co-ordinate on the road is the one where the line joining that co-ordinate and the house is perpendicular to the curve.

=> line is perpendicular to f(x) and passes through (0,0.5)

now slope of y=f(x)is y'=f'(x)=-1/2x^2

Let co-ordinate of point is (h,k)

so slope at (h,k)=-1/2h^2

=> slope of perpendicular line=-1/(-1/2h^2)=2h^2

=> line is:

y=2h^2*x+c

it passes through (0,0.5)

=> 0.5=2h^2*0+c

=> c=0

=> line is:

y=2h^2x

now solving this line with 1/2x,

2h^2*x=1/2x

Let distance be s. Let closest coordinate is (x,y).

=> s^2=(x-0)^2+(y-0.5)^2

=> s^2=x^2+(1/2x-0.5)^2

=> s^2=x^2+x^2/4+0.25-0.5x

=> s^2=5x^2/4-0.5x+0.25

As the coefficient of x^2 is positive, this function of x is an upward opening parabola, and therefore has a minimum rather than a maximum. This occurs where the derivative is 0:

5*2x/4-0.5=0

=> 2.5x=0.5

=> x=1/5=0.4

so y=1/0.4=0.25

so closest co-ordinate=(0.4,0.25)

b.)

distance of road=distance between (0.4,0.25) and (0,0.5)

=sqrt{(0.4-0)^2+(0.25-0.5)^2)}

=sqrt(0.16+0.0625)

=sqrt(0.2225)

=0.47 units

c.)

path of driveway => equation of line through (0.4,0.25) and (0,0.5)

=> slope=(0.25-0.5)/(0.4-0)=-0.25/0.4=-0.625

=> line is:

y-0.25=-0.625(x-0.4)

and y-0.5=-0.625(x-0)

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