Volume of a Box An open box is to be constructed from a piece of cardboard 20 cm by 40 cm by cut

Volume of a Box An open box is to be constructed from a...

Key Concepts

Optimization

Optimization involves finding the maximum or minimum value of a function under given constraints. In problems of this type, it typically means determining the variable value that yields the maximum volume or another optimal measurement, a common application in calculus and design problems.

Graphical Analysis

Graphical analysis uses visual representations, such as curves or plots, to understand the behavior of a function over its domain. It aids in estimating critical values like local or global maxima or minima by observing the shape, turning points, and overall trends of the function's graph.

Function Modeling

This concept involves converting a real-world situation into a mathematical function by identifying how a change in one variable affects another, such as constructing the volume function based on a variable dimension. It requires expressing the target quantity (like volume) in terms of one or more variables through a mathematical expression.

Domain of a Function

This concept pertains to identifying the set of all possible input values for which the function is defined and the outcomes are meaningful. In physical problems, this often involves imposing conditions so that all measurements, such as lengths, areas, or volumes, remain positive and realistic.

Transcript

00:01 Okay, so this question, we have an open box is to be constructed for a piece of cardboard 20 centimetres by 40 centimetres by cutting squares of side of x from each corner and folding up the sides as shown in figure a.

00:17 Express the volume v of the box as a function of x.

00:20 So that's part a.

00:20 Express the volume of v of the box as a function of x.

00:25 Okay.

00:26 So i've drawn a graph of the box here.

00:28 And you'll see.

00:30 See i've already put on some of the lengths and i'll just explain where i got these from.

00:37 So of course we've got our box was 20 centimeters by 40 centimeters.

00:44 Oh, don't know why that's just a one.

00:47 That was strange.

00:49 Just undo that real quick.

00:52 There we go.

00:52 So we got 20 centimeters by 40 centimeters.

00:55 And then we had corners cut out of length and width x.

01:01 So these are our corners here with length and width x.

01:04 And then i've got this 20 minus 2x here and this 40 minus 2x.

01:07 The way i got these is that i considered the full length and considered how much of the length has been cut to get these sides.

01:15 So the full length is 40 and we cut two xes.

01:19 So this length here and this length i'm highlighting here, this is 40 minus 2x and this width is 20 minus 2x.

01:27 And to find the volume, if you imagine folding this box together, our height is going to be x.

01:33 Our volume here is just going to be our height times our width times our volume.

01:37 Our width, imagine folding it upwards, is going to be 20 minus 2x.

01:44 And then our length is going to be 40 minus 2x.

01:50 I know a lot of people don't actually have great sort of imagination when it comes to sort of imagining how this looks as a box.

02:01 These lines may help.

02:02 You can imagine sort of folding these upwards.

02:06 It could even, in a test, you could even sort of like forward a piece of tracing paper or something, just sort of try and see what it looks like.

02:15 Yeah, it can be quite difficult and it is a bit unfair sometimes.

02:19 But if you can't see it, then you'll just have to use other various methods to try and see it.

02:26 There we go.

02:27 The volume is going to be x times by 20 minus 2x times by 40 minus 2x.

02:32 So that's our volume in terms of x.

02:34 And there we go.

02:35 That's the answer to part a.

02:36 For part b, it's asking you just to find a domain for v.

02:40 So what is the values that x can be? first off, i'm just going to draw a little graph of what v would look like.

02:48 Okay.

02:52 All right.

02:53 So we're going to have one turning point at zero here because of the x.