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Calculus/Tangent Lines

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Expert: Alon Mandes - 9/9/2009


Question
I believe this question is more pre-calculus, but I just can't seem to figure out where to even start attempting it. If you could outline the problem step-by-step that'd be great.

Question:
Find an equation of the straight line that is perpendicular to 5x-y=1 and is such that the area of the triangle formed by the positive x=axis, the positive y-axis, and the straight line is equal to 5.

Answer

drawing
I will call our tangent line T(x). We will suppose that T(x)=mx+n . & we want to find m & n.
1st, we know that T is perpendicular to y=5x-1, that means that : m*5=-1 . This gives us m :
m=-1/5 . Therefore T(x)=-x/5 +n . All we have 2 do now is to find n.
To do so, we will use the constrain :
"The area of the triangle formed by the positive x=axis, the positive y-axis, and the straight line is equal to 5". This, simply means that (xo*n)/2=5 . That's true because the triangle
formed is right one, & it's area is (xo*n)/2. Please see the drawing attached.
2nd, to find n, we have to find xo. To do so, let's substitute T=0 in the tangent line equation:
0=-xo/5 +n --> n=xo/5 --> xo=5n . Now, let's substitute this result into the area equation :
(5n*n)/2=5 --> 5n^2=10 --> n^2=2 --> n=0.707
So, our tangent line equation will be : T(x)=-x/5 +0.707

Alon.

Alon Mandes

Expertise

Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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M.A in Mathematics & Bs.c in Electronics.

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