Question

Help with optimization problem in calculus?

Answer 1

Express the length of the dumpster in terms of the width (x) of the bottom.

Width of bottom: x
Length of bottom: ?

Volume of dumpster (as a formula):

Volume of dumpster (as a numeral)

Depth of dumpster (can be derived from the above):

Answer 2

2x is length for the bottom

Answer 3

Your surface area function looks correct.
If you want to `minimize` that function, you'll need to look for a derivative, ya?

Answer 4

Or did you not fill in that blank?
I'm not sure :D

Answer 5

The bottom part is what I'm having problems with :(

Answer 6

The width, length and height

Answer 7

Take derivative of your SA function,
then set it equal to 0 to look for critical values.

You should be able to find an x value which corresponds to a minimum.
This will be your width.

Answer 8

4x-51/x^2

Answer 9

4x-(51/x^2)

Answer 10

x=2.34

Answer 11

Width is 2.34 yd?

Answer 12

Provided that your calculation of x is correct, yes.

Have you tried checking your x value?

does the product of length * width * height come out to 17 cubic yards?

What expression did you get for height?

length * width * height
4.68 * 2.34 * ? = 17 cubic yards?

I'd want to check that before moving on to calculate the minimum surface area.

Answer 13

Is it 1.55?

Answer 14

Brandon, I need to see your work ... I can't just look at 1.55 and say "right" or "wrong."

What was your expression for the depth of the dumpster? Type in your formula.

Answer 15

Much earlier, I typed the following to you:


Volume of dumpster (as a formula):

Volume of dumpster (as a numeral)

Depth of dumpster (can be derived from the above):

and you correctly answered that the length of the dumpster would be repr. by 2x
but you still haven't answered the other questions.

Answer 16

I just did 4.68*2.34 and got 10.9512 and then I did 17/10.9512 which is 1.55

Answer 17

Provided that x is actually 2.34, that's great. But how do you know 2.34 is correct?
I wanted to see how you got that.

Answer 18

Dumpster volume calculation:

(x) (2w) ( depth ) = 17 cu. yd.

depth = ? (type in a formula)

Answer 19

I checked it by multiplying the length and width and height and got 17

Answer 20

depth of dumpster? (formula)

Answer 21

2.34 times 2 = 4.68

Answer 22

No, we need a formula here, not a value.

My formula for the depth of the dumpster is 17 / (2x^2); do you agree?

Answer 23

L X W X H ?

Answer 24

Yes, later, but right now we're focusing on a formula for H.

Your H is ... what?

Answer 25

1.55

Answer 26

Is that a formula, Brandon?
Distinguish between numeric value and formula. The first is...obviously...a number. The 2nd may have constant coefficients, but is not a number, but a FORMULA.

Answer 27

depth (as a numeric value) 1.55, according to your calculations

depth (as a formula) 17 / (2x^2)

Please indicate whether you agree or disagree with thi8s formula.

Answer 28

I agree I meant to say that earlier

Answer 29

OK. Seems we're now on the same wavelength.

Now...the goal here is to find x so that the surface aera of the dumpster is a min.
We assume the dumpster is open. It has a bottom but no topi.

What is the area of one end of the dumpster? Type the formula (not a numeric value).

Answer 30

If you have the correct representations for the width of the end and the height of the end, then the formula for the area of the end should be a snap to write.

Answer 31

Width of end: x
Height of end: 17 / (2x^2)

Area of end:

Answer 32

Do you do width times height?

Answer 33

Yes. Area of a rectangle = width times height.

Answer 34

3.627

Answer 35

brandon, we are talking about formulas, NOT numeric values.

Why are we doing this? becasue we want to minimize the area of the dumpster. To do that, we must have a formula in terms of x (NOT in terms of numeric values) to differentiate.

What is the formula (not the value) for the area of the end of the dumpster?

It's length time s width. Find the formulas for the width and the length above.

Answer 36

The formula for the area of ONE end of the dumpster is

width * height = area
x 17/(2x^2) = ???

Answer 37

Width was 4x-(51/x^2)

Answer 38

Length is 4x-(51/x^2) times 2

Answer 39

No, the width of the end is simply x.
The height of the end is 17 / (2x^2).

Find the product of these 2 quantities.

Answer 40

Length is 2x

Answer 41

Refer to the original illustration.
It shows that the width of the dumpster (and the width of the end of the dumpster) is x.
Then the length of the dumpster is 2x. YES.

But what is the height of the end of the dumpster? We have discussed this before.

Answer 42

1.55?

Answer 43

Note that the illustration shows a ' ? ' for the height.

No, Brandon, the height of the end is a variable quantity, not a number.

Go back and review this discussion; you'll see that the height of the end of the dumpster is 17 / (2x^2).

You MUST be comfortable with this before we move on.

Answer 44

is it just "H"?

Answer 45

No, it's H = 17 / (2x^2).

If necessary we could again go through the work necessary to find that formula 17 / (2x^2), but we've been through it several times already.

Answer 46

Let's do it again any way.

Volume of dumpster is length * width * depth = 17
or 2x x depth =17
or 2x^2 * depth = 17

Solve this for depth, please.

Answer 47

2.04?

Answer 48

No numbers, please. We won't be dealing with numbers until the very end of this problem.