Answer to Question #243502 in Math for N1yxz

Question

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.

Answer 1

Let $y_1=ax+b, y_2=mx+n.$

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

$$y_2(0)=n=3$$

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

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

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

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

If $y_1=y_2,$ then

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

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

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

$$y_1=-\dfrac{8}{7}(\dfrac{35}{12})+8=\dfrac{14}{3}=y_2$$

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

$$d=19.44$$