A rectangle is to be inscribed in a semicircle of radius 5...

A rectangle is to be inscribed in a semicircle of radius 5 $\mathrm{cm}$ as shown in the figure. (a) Show that the area of the rectangle is modeled by the function $$A(\theta)=25 \sin 2 \theta$$ (b) Find the largest possible area for such an inscribed rectangle. (c) Find the dimensions of the inscribed rectangle with the largest possible area.

Transcript

00:01 So in this problem, we're given the semicircle with a rectangle inscribed in it as such.

00:08 And the semicircle has a radius of 5 centimeters.

00:12 And we're first asked to show that the area a of the rectangle can be modeled as the function a of theta equals 25 times the sine of to theta.

00:40 All right.

00:42 Well, we first, if this is theta right here, okay, well, then what do we know? well, let's see, the area of a rectangle, right? the area of a rectangle is the length times the width.

01:00 Okay.

01:01 Well, we know that the length, right, will be this total distance right here.

01:20 Which will be two times that distance.

01:29 I'm going to call that distance d here and d here, right? so that'll be two times d.

01:40 Okay.

01:42 And we also know that, because of angle theta, that d is a cosine of theta, actually d over 5, right, because the cosine of the angle is the adjacent side over the hypotenuse, which means then that d is 5 times the cosine of theta.

02:28 Okay.

02:30 So then that means that the length is 10 cosine of theta.

02:38 Okay.

02:40 What about the width? well, the width is that height's right there, isn't it? okay.

02:47 And we know that sine of theta is that opposite side, which is that width over the hypotenuse, which is five.

02:57 So this gives me then that the width is five times the sign of theta.

03:05 Okay...