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Expert: - 6/3/2009


Question

Graph
Hello, I was just wondering if you could help me find the
solution to this problem:

For each x in the interval [0, 3], consider the triangle
with vertices (0, 0), (x, 0) and (x, p(x)) where p(x) = 3
+ 2x − x^2. Find the value of x which generates the
triangle with the largest area.

All assistance would be greatly appreciated. Thanks.

Answer
Hi Jennifer,
Ok, so let's see if i can help.
(0,0) is the origin and the point (x,0) lies on the x axis at a distance of x units from the origin. The point (x,p(x)) is a point on the curve p(x) corresponding to x. Sketching the triangle you find that the base is the line from the origin to (x,0) and the perpendicular height has a value of p(x) units.
The area of a triangle is half its base x perpendicular height,
A = 1/2.x.p(x)
 = 1/2.x.(3 + 2x - x^2)
 = (3x + 2x^2 - x^3)/2
The area is largest when dA/dx = 0 and the second differential is negative
dA/dx = (3 + 4x - 3x^2)/2
equating to zero
(3 + 4x - 3x^2)/2 = 0
3 + 4x - 3x^2 = 0
x = -4 ± sqrt[16 - (4)(-3)(3)] / 2(-3)
 = [-4 ± sqrt(52)]/2
 = -0.54 or 1.87
One of this gives a maximum area and the other gives a minimum area. The value that gives the maximum area is the one that gives a negative second differential i.e d/dx(dA/dx)
Now,
d/dx(dA/dx) = (4 - 6x)/2
           = 2 - 3x
at x = -0.54
2 - 3x = 0.38
at x = 1.87
2 - 3x = -3.61
and so, the value of x which generates the triangle with the largest area is x = 1.87

Regards

Ahmed Salami

Expertise

I can provide good answers to questions dealing in almost all of mathematics especially from A`Level downwards. I can as well help a good deal in Physics with most emphasis directed towards mechanics.

Experience

An engineering graduate. I have been doing maths and physics all my life.

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