Greatest possible rectangle

Calculus Level 4

A rectangle has one of its bases on $y=0$ and is contained between $x=3$ and $x=-1$ . It's opposite base has two of its points on the graph of $y=x^3-2x^2+1$ with $y>0$ . Find the perimeter of the rectangle with the greatest possible area that fits the above definition.

Adopted from one of Michael Mendrin 's problems .

The answer is 6.

3 solutions

Kumarmanas Nethil
Jul 21, 2014

The two points mentioned needn't necessarily be the end points. In that case, the max area will be 4*1=4, and consequentially, the perimeter will be 10.

Vaibhav Borale
Jul 22, 2014

Plotting the graph of equation y=x^3-2x^2+1 , we get 2 peaks wise at points (0,1) & (3,10). Now, I chose the value 1 for y and got 2 solutions of the same equation and hence 2 distinct point which lie on the graph wise (0,1) & (2,1) , these are nothing but the required points of the extremities of the another base.

So, finally we got all 4 points of vertices of rectangle i.e. (0,1), (2,1), (0,0) & (2,0) giving us the perimeter 2(l+b)=2(2+1)=6

It says that the base line is y=0 and is contained between x=3 and x=-1 which means those are the extremities of the bases. Hence the width of the rectangle is already determined as 4. Now, we need to find the opposite base that is parallel to this one. It also says that two of the points on the opp base is on the cubic curve given. IT DOES NOT SAY that those two points need to be the extremities of the opposite base. Hence your assumption is wrong there. The two peaks cut the line y=0 at x=0 and x=2. These two points lie on the base and the curve both. Hence, the opposite base is the line y=1 contained between x=-1 and x=3. The perimeter should hence be P= (4+1)*2= 10. Please see if I'm correct.

Arnab Bhattacharya - 6 years, 9 months ago

Rather than graphing it, you could take the derivative and find the roots of the derivative.

Trevor Arashiro - 6 years, 10 months ago

I approached that way also....but as we are given limit boundaries of x also, it was bit confusing. And again even after finding one of the extremities by derivative method you have to find corresponding base point again which would have been lengthy......then I chose this method.

VAIBHAV borale - 6 years, 10 months ago

Trevor Arashiro
Jul 17, 2014

To find the maximum area of a rectangle contained by a graph, you need to find the relative maxima of the graph in this equation. Using the first derivative, y=-4x^3+2x, we need to find the roots of this equation to find relative max values of the original equations. Doing some calculations, we find the max values to occur at x=√(2)/2 and x=-√(2)/2.

Adding these together, we find the length of the top of the rectangle to be √(2). Plugging in x=√(2)/2 into the original equation, we find the max value to be y=1/4. Multiplying the height by the width, we find the area to be √(2)/4. Thus a and b are 2 and 4, so our answer is 6.

Greatest possible rectangle

Adopted from one of Michael Mendrin 's problems .

The answer is 6.

3 solutions

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