<p>I was Wondering if some of you could help Me with a Precalculus mathematics problems I seem to be stuck on.</p>
<p>The question is as follows:</p>
<p>A rectangular tunnel is cut through a mountain to make a road. The upper vertices of the rectangle are on the circle x^2 + y^2 = 400, as illustrated in this figure:</p>
<p>(a) show that the cross sectional area is 400 sin2z (where “z” is the angle of the cross-section…I can’t make that “O”-like shape in the picture)</p>
<p>b) find the dimensions of the rectangular end of the tunnel that maximizes its cross-sectional area.</p>
<p>thank you folks. Usually I have no trouble with this sort of problem, but I seem to be stuck! :(</p>
<p>by the equation given, we know the radius of the semicircle is 20. thus, our handy right triangle has hypotenuse 20. Then, we use the triangle again to find that the height of the rectangle is 20sinθ, and the width of the rectangle is 2*20cosθ. (note: we must have the factor of 2 because the base of our right triangle is only half the width we want.)</p>
<p>multiply em together and we have area = 800sinθcosθ. since 2sinθcosθ = sin2θ, we have area = 400sin2θ as desired. </p>
<p>Then, we know the range of 400sin2θ is [-400, 400] because the range of sin2θ is [-1, 1]. So the maximum area must be 400.</p>
<p>since the domain for our theta is [0, π/2], the only value of θ that works is π/4. plugging into our height/width expression, we find the dimensions of of the rectangle is 10<em>sqrt(2) by 20</em>sqrt(2).</p>
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