Answer to Question #243502 in Math for N1yxz

Question #243502

A civil engineer found that the durability dd of the road, she is laying depends on two functions y1y1 and y2y2 as follows: d=y22+4y1−21d=y22+4y1−21. Functions y1y1 and y2y2 depend on the amount of plastic (xx) mixed in bitumen, and their variations are linear functions of xx. Let y1=8y1=8 and y2=3y2=3 for x=0x=0 and y1=0y1=0 and y2=7y2=7 for x=7x=7. Find the durability of the road (upto 2 decimal points), if the amount of plastic is such that both the functions are equal.


1
Expert's answer
2021-09-29T09:26:08-0400

Let y1=ax+b,y2=mx+n.y_1=ax+b, y_2=mx+n.


y1(0)=b=8,y_1(0)=b=8,

y2(0)=n=3y_2(0)=n=3

y1(7)=a(7)+8=0=>a=87y_1(7)=a(7)+8=0=>a=-\dfrac{8}{7}

y2(7)=m(7)+3=7=>m=47y_2(7)=m(7)+3=7=>m=\dfrac{4}{7}

y1(x)=87x+8y_1(x)=-\dfrac{8}{7}x+8

y2(x)=47x+3y_2(x)=\dfrac{4}{7}x+3

If y1=y2,y_1=y_2, then


87x+8=47x+3-\dfrac{8}{7}x+8=\dfrac{4}{7}x+3

127x=5\dfrac{12}{7}x=5

x=3512x=\dfrac{35}{12}

y1=87(3512)+8=143=y2y_1=-\dfrac{8}{7}(\dfrac{35}{12})+8=\dfrac{14}{3}=y_2

d=(143)2+4(143)21=175919.44d=(\dfrac{14}{3})^2+4(\dfrac{14}{3})−21=\dfrac{175}{9}\approx19.44

d=19.44d=19.44


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