The arch of a bridge forms a parabola with the function \displaystyle f(x)=-0.05(x-30)^{2}+45f(x)=−0.05(x −30) 2 +45 where x is the horizontal distance (in meters) from the arch's left end and y is the height (in meters) above the road. How many meters high is the arch at its highest point?

Step-by-step explanation:

If the arch of a bridge forms a parabola with the function

f(x)=−0.05(x−30)² +45 where x is the horizontal distance;

The arc is at its highest point at when df(x)/dx = 0

df(x)/dx  = -2(0.05)(x-30)(1) + 0

df(x)/dx = -0.1(x-30)

df(x)/dx  = -0.1x+3

Since df(x)/dx  = 0

-0.1x+3 = 0

-0.1x = -3

x = -3/-0.1

x = 20

Substitute x = 20 into the function as shown:

f(x)=−0.05(x−30)² +45

f(20)=−0.05(20−30)² +45

f(20)=−0.05(-10)² +45

f(20)=−0.05(100) +45

f(20)=-5+45

f(20) =  40

Hence the arch is 40meters high at its highest point