Question

I Consumer Economics Suppose your car contains just one gallon of gas. Driv-
ing at 20 mi/h you can go 26 mi. Likewise, you can go 34 mi driving at
40 mi/h and 32 mi driving at 50 mi/h.
a. Find a quadratic function that models this data.
b. How far could you go if you drove at 65 mi/h?
c. The nearest gas station is 16 mi away. If the speed limit is 55 mi/h, at what
maximum speed could you drive and still reach it?

Answer 1

Answer:

See below

Step-by-step explanation:

A) The quadratic function that models this data is $f(x)=-0.02x^2+1.6x+2$ which you can view in the graph attached (done by regression).

B) $f(65)=-0.02(65)^2+1.6(65)+2=21.5$, or 21.5 miles

C) Set $f(x)=16$ and solve for x:

$f(x)=-0.02x^2+1.6x+2$

$16=-0.02x^2+1.6x+2$

$0=-0.02x^2+1.6x-14$

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{-1.6\pm\sqrt{(1.6)^2-4(-0.02)(-14)}}{2(-0.02)}$

$x=\frac{-1.6\pm\sqrt{2.56-1.12}}{-0.04}$

$x=\frac{-1.6\pm\sqrt{1.44}}{-0.04}$

$x=\frac{-1.6\pm1.2}{-0.04}$

$x_1=\frac{-1.6+1.2}{-0.04}$

$x_1=\frac{-0.4}{-0.04}$

$x_1=10<55$

$x_2=\frac{-1.6-1.2}{-0.04}$

$x_2=\frac{-2.8}{-0.04}$

$x_2=70>55$

Since $70>55$, then 10mph is the maximum speed you could drive and still reach the gas station that is 16 miles away when the speed limit is 55.