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The diagram shows the graph of y=4-3x^2, for x≥0 and y≥0. B is a point on the graph and OABC is a rectangle. Find the value of x for which the area of OABC is a maximum.

 Aug 28, 2019
 #1
avatar+23884 
+3

Applications of differentiation and antidifferentiation

 

The diagram shows the graph of y=4-3x^2, for x=0 and y=0.

B is a point on the graph and OABC is a rectangle.
Find the value of x for which the area of OABC is a maximum.

 

...here is no diagram.

 

I assume:

 

\(\text{Let $\mathbf{A}$ is the area of OABC }\)

 

\(\begin{array}{|rcll|} \hline A &=& x\cdot y \\ &=& x\cdot (4-3x^2) \\ &=& 4x-3x^3 \\\\ A' &=& 4-9x^2 \quad | \quad A' = 0 \\ 4-9x^2 &=& 0 \\ 9x^2 &=& 4 \quad | \quad \text{square root both sides} \\ 3x &=& 2 \\\\ \mathbf{x} &=& \mathbf{\dfrac{2}{3}} \quad | \quad A''=-18x =-18\cdot \dfrac{2}{3}=-12\ (A''<0 \text{ maximum}) \\ \hline \end{array}\)

 

laugh

 Aug 28, 2019
edited by heureka  Aug 28, 2019
edited by heureka  Aug 28, 2019
edited by heureka  Aug 28, 2019
edited by heureka  Aug 29, 2019
 #2
avatar+106535 
+3

Thanks, heureka.....

 

Here is the graph  of the max rectangle :

 

 

 

cool cool cool

 Aug 28, 2019
 #3
avatar+23884 
+2

Thank you, CPhill !

 

laugh

heureka  Aug 29, 2019

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