calc word problems *worksheet* help?

[whinny.s18]

I was doing great until i reached this problem on my calc worksheet. think anyone can help me?

an electrical power line hung between two poles models has the mathematical equation F(x)= 1/180x^2- 1/3x +20 and 0(less than or equal to)x(less than or equal to) 60. where x represents the horizontal distance and f(x) is the height in feet.
that all was pretty easy to under stand.
a. What is the location (x,y) f the lowest point of the wire? - for this i tried to do the x,y intercepts. but that didnt seem right. am i supose to graph them?
then the next one is
b. draw a picture of the wire through out its entire domain. - after i didnt under stand the first part of this question i figured that this part woudnt be able to be answered.

if anyone can help me with this i would be forever thankful of them.

 

[BigGlenntheHeavy]

\(\displaystyle F(x) \ = \ \frac{x^2}{180}-\frac{x}{3}+20, \ 0 \ \le \ x \ \le \ 60.\)

\(\displaystyle F'(x) \ = \ \frac{x}{90}-\frac{1}{3}, \ setting \ to \ zero \ yields \ x \ = \ 30.\)

\(\displaystyle Hence, \ F(30) \ = \ 15 \ = \ absolute \ minimum.\)

 

[galactus]

They're modeling the wire over a parabola. (I say this because they are usually modeled over a 'catenary', which at first glance looks like a parabola).

Using \(\displaystyle x=\frac{-b}{2a}, \;\ y=c-\frac{b^{2}}{4a}\), the coordinates of the vertex can be found. This is the lowest point.

Let a=1/180, b=-1/3, c=20 and plug them in.

You can easily graph this parabola on a calculator.

 

[whinny.s18]

thank you so much! so would i use the -b/2a for problems asking for the maximum of something? and also how did you come up with the y=c-b^2/4a? is that universal for all y coordinates?

 

[galactus]

Yes, it is. If you plug \(\displaystyle x=\frac{-b}{2a}\) back into \(\displaystyle y=ax^{2}+bx+c\), that is what you get for y.

 

[whinny.s18]

awesome. i understand the problem now. thank you.