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A triangle is formed with one vertex at the vertex of the parabola y=x^2-1 and the other two vertices at the intersections of the line y=r and the parabola. If the area of the triangle is between 8 and 64 inclusive, find all possible values of r. Express your answer in interval notation.

 Dec 31, 2019
 #1
avatar+109563 
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By the graph here : https://www.desmos.com/calculator/zascepyruy

 

Note that :

 

The height  of the triangle  =   x^2

And the base of the triangle  =  2x

 

So   we have  that

 

(1/2) x^2  * ( 2x )   =  8

x^3  =  8

x = 2

So the minimum possible height =  2^2  = 4

So   y   =  3

 

And

(1/2) x^2  * (2x)  =  64

x^3  =  64

x  = 4

So....the max possible height =  4^2  = 16

So  y =  15

 

So

3  ≤  r  ≤  15

 

cool cool cool

 Jan 1, 2020
 #2
avatar+287 
0

Thank you, I have a better understanding of what this problem meant. I first tried plugging in some values and then graphing it. I figured there was an easier way!

 Jan 1, 2020

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