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In the following exercises, solve the given maximum and minimum problems.An architect designs a window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is to be $6.00 \mathrm{m},$ what dimensions of the rectangle give the window the largest area?

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 7

Applied Maximum and Minimum Problems

Derivatives

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Okay, so we have a triangle with the sides. Um, X equals zero. Why equals zero and X, divided by four. Plus why divided by six is equal to one. All right, this has, um, a, uh, intercept here at the 0.40 Right, cause if y equals zero all right. Y equals zero X must then equal to four. Right? And then this also has an intercept here at six. Sorry. Zero zero comma six zero comma. Six. Where if What are Sorry if X equals zero, then why must equal six? Right? So we're trying to optimize the area of a rectangle that fits inside. Uh, this this triangle. Right? So the formula for this rectangle is essentially going to be a is equal to x times. Why? Where x and y are going to fall on the line. So we're gonna have X divided by four. Plus why divided by six is equal to one, um, multiplying by 24 on both sides. For the for the constraint, we get something a little simpler. Six x plus four. Why is equal to 24 So solving for why we get that why is able to 1/4 times 24 minus six x Talking this into our formula for the area we got. That area is equal to X times. Ah, All right. Time's 1/4 24 minus six x or in total area. You see the two 1/4 for simplifying six X squared. Like so. So now we need to find the derivative. So rewriting the area is equal to 1/4. 1/4 24 X minus six X squared. Right. So we need to find the derivative of the area with respect to X. This is going to be 1/4 times 24 minus 12 x. Remember, we're sending that equals zero. So we get that 24 minus 12 x is equal to zero or 12 X is equal to 24. So ex is equal to two now, Um, why here then? Um, we plug in. If you look back here, uh, why is equal to 1/4? 1/4 24 minus six x right, 24 minus six x. So why is going to be equal to 24 minus six times two. So 1/4 of, uh, 12. So this is going to be three in the area. It's gonna be two times three. So our area is going to be six. So this is our answer. Uh, here six.

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