The trough in the figure is to be made to the dimensions shown. Only the angle θcan be varied. W

The trough in the figure is to be made to the dimensions...

Key Concepts

Geometric Modeling

Geometric modeling is the process of translating a physical scenario or design into a mathematical framework. This involves identifying relevant dimensions and relationships—in this case related to volume and angles—and expressing them as mathematical functions that can be analyzed and optimized using techniques like differentiation.

Calculus-based Optimization

This concept involves applying calculus techniques, especially differentiation, to find the maximum or minimum values of a function. In optimization problems, you typically set the derivative equal to zero to find critical points, which may correspond to maximum or minimum values of the function under consideration.

Trigonometric Function Optimization

Many optimization problems, particularly those involving angles, rely on trigonometric functions such as sine and cosine. Optimizing such functions involves understanding how variations in the angle affect the overall function and applying differentiation techniques to find the angle that yields an optimal (maximum or minimum) value.

Transcript

00:01 Okay, what we are going to be doing is being able to maximize a volume of a trough.

00:12 And the trough, see if i can draw them out, is made up of, so here is the trough.

00:39 And so there is the trough that is 20 feet long.

00:57 And so we also know that the in caps, the in caps.

01:06 So this area right here, if i redraw that area right there, it's a trapezoid shape.

01:17 It is one foot, one of the bases, or the bottom.

01:23 Base is one foot and what we want to do is determine the value of theta to maximize the volume of the trough.

01:53 Okay, so now we've got to come up with a few things.

01:57 We're also told that those dimensions are one foot as well.

02:14 And so we know that the volume of a trough is going to be given by the area of, i'm going to use these trapezoids as my basis.

02:34 So it's going to be the area of those bases times this length.

02:40 It's going to give me my volume.

02:42 And so that's going to be 1 half b1 plus b2 times h is the area of my trapezoid times that 20.

03:00 So this is going to be equal to 10 times.

03:10 B1 plus b2 times h now we got to figure out what is b1 what is b2 and what is h okay so and this is where we are going to and then we have to also bring in theta and so we also know that this is one inch so if i let this be x right there then we also know that x is opposite of that angle.

03:48 And so we can bring in, of course, trigonometry.

03:52 So x is equal to sine of theta because my hypotenuse is one.

04:06 And if i let this be y, we also know that y is equal to kose.

04:19 Sine theta.

04:21 Well, what we also know is that b1 is equal to one foot.

04:33 B2 is equal to 1 plus 2x.

04:42 And h is y.

04:47 And so we can actually substitute those in.

04:51 So my volume is equal to 10.

04:55 Times b1, which is 1, plus b2, which is 2, which is 2 times sine of theta, times my height, which is going to be cosine theta.

05:16 And so there i have my volume formula.

05:19 And so what we're going to do now is to take the dream.