Can someone please help me with this question.

Question

The Fuel efficiency F for a particular  car is given by

F(x) = -0.015x² + 1.35x + 7.8

where x is the speed of the car in miles per hour (mph) and F(x) is the corresponding fuel efficiency in miles per gallon(mpg).

 

a) What is the fuel efficiency if the car's speed is 10 mph?

 

b) What speed will yield the maximum fuel efficiency?

 

So when i attempted to solve this equation I did not understand how to find b but for part a this is what I did...

 

f(x)=-0.015x² + 1.35x +7.8

f(x) = -0.015(10)² + 1.35 (10) + 7.8

f(x) = - 0.015(100) + 1.35(10) + 7.8

f(x) = -1.5+(1.35)(10) +7.8

f(x) = -1.5 + 13.5 + 7.8

f(x) = 12 + 7.8

f(x) = 19.8

 

I am not sure if I would just put 19.8 mpg or if I would round this to 20 mpg. Again as for the part b i am not sure what my next step would be.

 

Answer 1

Perfect; first part is done.

 

For next part, you have to find a extrema  of function f(X). Just derife the f(x) and equalize it to zero, ok?

 

f(x)=-0.015x2 + 1.35x +7.8

f'(x)= -0.03x+1.35 = 0

x=1.35/0.03= 45 (mph)

Answer 2

Go with 19.8 because that's the most accurate, and notice that the constant in the formula for F has one decimal place, so that's a good idea of what accuracy is possible &/or desired.

 

Part b: notice that the function's graph is an upside-down parabola.  You can find the maximum value by evaluating the function at its vertex; the formula for x_vertex is -b/(2a) where a is the coefficient of x squared, b is the coefficient of x.