y=-x^2+10x+30 where x and y are measured in feet. determine the width. determine the height.

Question

the intersection of a small hill is modeled by the equation y=-x^2+10x+30 where x and y are measured in feet.

Answer 1

Alright, so we have the equation y=-x^2+10x+30, which is a quadratic equation (parabola) but the -x^2 indicates that it is upside down.  So we have a graph of exactly what they describe (a 'hill').  In this instance, the x-axis serves as the "ground" where the width will be measured, and the distance from the vertex (center) down to the x-axis will be the height.

 

So lets start with the width.  To solve for the width, we need to know the distance between both x-intercepts of the parabola.  So what values of x will make it so that y = 0?  There isn't an easy way to solve this so we're going to have to go with the quadratic equation.  We let a = -1, b = 10, and c = 30.  Plugging these into the quadratic equation we get:

 

(-10 +/- sqrt(10^2 - 4(-1)(30))) / -2

 

Simplified:

 

(-10+/- sqrt(100+120)) / -2

(-10 +/- sqrt(220)) / -2

We can remove sqrt(4) from sqrt(220) to get 2(sqrt(55)) thus simplifying to:

(-10 +/- 2(sqrt(55))) / -2

5 +/- sqrt(55)

 

So, the x-coordinates for our x-intercepts are where x = 5 +/- sqrt(55).  So 5 is the center coordinate of the hill, and to either side of it stretches exactly sqrt(55) ft.  So the width will be exactly 2 times that, so width = 2 * sqrt(55).

 

For the height, we know the center of the hill is the tallest, and we found where the center was in the previous step (it's where x = 5), so we simply need to plug in x = 5 into the original equation.  This gets us:

 

y = -(5^2) + 10(5) + 30

y = -25 + 50 + 30

y = 55

 

So the y coordinate of the center of the parabola (the "top" of the hill) is 55, making the height 55 feet tall.

 

I hope this helps.

 

Answer 2

We need more information, since there is only 1 equation, but there are 2 variables!