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A rectangle that is \[x\] feet wide is inscribed in a circle of radius \[10\] feet. How do you find the expression for the area of the rectangle as a function of \[x\]?

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Last updated date: 25th Feb 2024
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IVSAT 2024
Answer
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Hint: From the question given, we have been given that a rectangle that is \[x\] feet wide is inscribed in a circle of radius \[10\] feet. And we have been asked to find the expression for the area of the rectangle as a function of \[x\]. We can solve the given question by reading the question thoroughly and by understanding it we can get the answer. Let us assume the length as $y$

Complete step by step answer:
From the question, we have already given that a rectangle that is \[x\]feet wide is inscribed in a circle of radius \[10\] feet.
Let us assume that the length of the rectangle is $y$
From this statement given in the question, we can understand that the diagonal of the rectangle is the diameter of the circle (which is twice the radius).
Pythagoras theorem is stated as “in a right angle triangle, the square of the longest side that is the hypotenuse is equal to the sum of the squares of the other two adjacent sides”.
Therefore, by using the Pythagoras theorem to the above question \[\Rightarrow {{20}^{2}}={{x}^{2}}+{{y}^{2}}\]
Now, simplify further to get the expression for the area of the rectangle as a function of \[x\].
By simplifying further, we get \[\Rightarrow 400={{x}^{2}}+{{y}^{2}}\]
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Shift \[{{x}^{2}}\] to the left hand side of the equation and rearrange it.
By shifting \[{{x}^{2}}\] to the left hand side of the equation and rearranging it, we get
\[\Rightarrow {{y}^{2}}=400-{{x}^{2}}\]
$\Rightarrow y=\sqrt{400-{{x}^{2}}}$
As of now we have the value of $y$ .
From the basic concept we know that the area of any rectangle is given by the product of its length and breadth.
Mathematically given as $Area=xy$
$\Rightarrow Area=x\left( \sqrt{400-{{x}^{2}}} \right)$
Hence, we got the expression for the area of the rectangle as a function of \[x\].

Note: We should be very careful while solving this type of questions. Also, in this type of question we should have to read the question thoroughly and understand it. Then the question can be solved easily. Also, we should be very careful while doing the calculation. We should make a note that here we have been asked for a rectangle not square if for square then all sides will be equal so $x=y$ therefore $2{{x}^{2}}=400\Rightarrow x=10\sqrt{2}$ then area will be $200$ .