# A rectangle is to have an area of 16 square inches. How do you find its dimensions so that the distance from one corner to the midpoint of a nonadjacent side is a minimum?

We can write the following equations:

Draw a diagram of the line cutting through the rectangle and use the Pythagorean Theorem to say that the length of the segment can be found through:

Thus,

Simplify:

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To find the dimensions of the rectangle so that the distance from one corner to the midpoint of a nonadjacent side is minimized, we need to use calculus. Let the dimensions of the rectangle be (x) and (y), with (x) being the length and (y) being the width. The area of the rectangle is (A = xy = 16). The distance from one corner to the midpoint of the nonadjacent side can be represented by the function (d(x) = \sqrt{x^2 + \left(\frac{y}{2}\right)^2}).

To minimize (d(x)), we first express it solely in terms of one variable. Since (A = 16), we have (y = \frac{16}{x}). Substitute this into (d(x)) to obtain (d(x) = \sqrt{x^2 + \left(\frac{8}{x}\right)^2}).

Next, we differentiate (d(x)) with respect to (x) and set the derivative equal to zero to find critical points:

[ \frac{d}{dx} d(x) = \frac{x}{\sqrt{x^2 + \left(\frac{8}{x}\right)^2}} - \frac{8}{x^2}\left(\frac{8}{x}\right) = 0 ]

Solving this equation gives us the critical points. After finding the critical points, we check for local minimums by using the second derivative test.

Finally, we check the boundary points of the feasible domain (0 < x < \infty) to ensure that our solution is indeed the minimum.

After finding the value of (x), we can find the corresponding value of (y) using (y = \frac{16}{x}). These values of (x) and (y) will give the dimensions of the rectangle that minimize the distance from one corner to the midpoint of a nonadjacent side.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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