Need help on a geometric/ alegba question please help.

[niham]

A juice factory needs to build a tank capable of holding 800πm³ of juice. The tank will be cylindrical with the circular sides on the top and bottom. The metal needed to make the curved surface of the tank will cost $32/m², while the top and bottom will be made of lower quality metal which costs $25/m². We wish to find the value of r and h which minimize the cost of the tank.

For this question you may need the following formulas. The volume of a cylinder is V=πr²h. The area of the top or bottom of a cylinder(area of a circle) is A=πr², and the area of the curved surface of a cylinder is A=2πrh.

a) Find the formula for the total cost of the tank in terms of the radius r and the height h.

b) Use the given value for the volume of the tank to find a formula for h in terms of r.

c) Plug the answer for part (b) into the answer for part (a) to get a formula for the total cost in terms of the variable r.

d) Use a graphing calculator to find the value of r which minimizes the total cost. (Hint: use the window Xmin=-10, Xmax=10, Ymin=0, Ymax=2000)

e) Use the answer to part (d) to get the value of h which minimizes the total cost.

What I have so far is: 800nm^3 --> 800x10^-12m^3
a) I get this formula
a) Total cost = ($32/m²)(2πrh) + ($25/m²)(2πr²) but I don't know how to get radius
b) I rearranged v=(π)(r)(h) to get h = (800 x 10^-12m³)/(πr²)
but for d and e I don't really get how to do it.

 

[ksdhart2]

What are your thoughts? What have you tried? Please share with us any and all work you've done on this problem, even if you know it's wrong. Thank you.

 

[niham]

this is what I have so far

What are your thoughts? What have you tried? Please share with us any and all work you've done on this problem, even if you know it's wrong. Thank you.

800nm^3 --> 800x10^-12m^3
Hi, so for a) I get this formula
a) [FONT=&quot]Total cost = ($32/m²)(2πrh) + ($25/m²)(2πr²) but I don't know how to get radius
b) I rearranged v=([/FONT][FONT=&quot]π[/FONT][FONT=&quot])(r)(h) to get [/FONT][FONT=&quot]h = (800 x 10[/FONT]^-12[FONT=&quot]m[/FONT]³[FONT=&quot])/(πr[/FONT]²[FONT=&quot])
but for d and e I dont really get how to do it. [/FONT]

 

[Harry_the_cat]

A juice factory needs to build a tank capable of holding 800πm³ of juice. The tank will be cylindrical with the circular sides on the top and bottom. The metal needed to make the curved surface of the tank will cost $32/m², while the top and bottom will be made of lower quality metal which costs $25/m². We wish to find the value of r and h which minimize the cost of the tank.

For this question you may need the following formulas. The volume of a cylinder is V=πr²h. The area of the top or bottom of a cylinder(area of a circle) is A=πr², and the area of the curved surface of a cylinder is A=2πrh.

a) Find the formula for the total cost of the tank in terms of the radius r and the height h.

b) Use the given value for the volume of the tank to find a formula for h in terms of r.

c) Plug the answer for part (b) into the answer for part (a) to get a formula for the total cost in terms of the variable r.

d) Use a graphing calculator to find the value of r which minimizes the total cost. (Hint: use the window Xmin=-10, Xmax=10, Ymin=0, Ymax=2000)

e) Use the answer to part (d) to get the value of h which minimizes the total cost.

What I have so far is: 800nm^3 --> 800x10^-12m^3
a) I get this formula
a) Total cost = ($32/m²)(2πrh) + ($25/m²)(2πr²) but I don't know how to get radius
b) I rearranged v=(π)(r)(h) to get h = (800 x 10^-12m³)/(πr²)
but for d and e I don't really get how to do it.

800πm³ ..... is this \(\displaystyle 800\pi\) sq metres?

 

[niham]

800πm³ ..... is this \(\displaystyle 800\pi\) sq metres?

yes sorry, just a typo it is 800pim^3

 

[ksdhart2]

Okay. First off, thank you for sharing your work. I see so many people on this forum who leave immediately and never return upon being asked to try, even a little bit, that's it's refreshing to see a student whose not looking for free answers.

800nm^3 --> 800x10^-12m^3

This would be true, if the volume of the tank were measured in nanometers. But I don't think it applies here, as a tank with a volume of 800 cubic nanometers would be very very small (for comparison, 800 nm is approximately the size of the "giant" virus Mimivirus). I think what the problem meant was that the volume of the tank is 800 pi cubic meters: \(\displaystyle 800 \cdot \pi \text{ m}^3\). You may want to double check with your instructor to be 100% sure though.

a) Total cost = ($32/m²)(2πrh) + ($25/m²)(2πr²)

I agree with your workings on this step, with the exception that you need to label your units. The surface areas of curved section of the cylinder and the top and bottom of the cylinders are measured in square meters. So you have:

\(\displaystyle \left( \dfrac{$32}{m^2} \right) (2 \pi rh \: m^2) + \left( \dfrac{$25}{m^2} \right) (2 \pi r^2 \: m^2)=$64 \pi rh + $50 \pi r^2\)

b) I rearranged v=(π)(r)(h) to get h = (800 x 10^-12m³)/(πr²)

If my earlier assumption is correct, then I also agree with your workings here, once the appropriate substitution of the volume actually being 800 pi cubic meters instead of 800 cubic nanometers is made. Be sure to also label your units here too though.

c) Plug the answer for part (b) into the answer for part (a) to get a formula for the total cost in terms of the variable r.

So for this part, what happens if you follow the directions? If you use your formula from part (a), let total cost be defined by the variable c, and replace ("plug in") every instance of h with the formula you found in part (b), then simplify, what do you get?

d) Use a graphing calculator to find the value of r which minimizes the total cost. (Hint: use the window Xmin=-10, Xmax=10, Ymin=0, Ymax=2000)

The answer to part (c) will be a formula in the form c = (some function of r). This is what the problem is asking you to graph. If your calculator is anything like mine, it can only graph y as a function of x. But you can work around that by letting y = c and x = r. When you graph the function, what do you see? Approximately where is the minimum value? Is it a "nice" integer value?

e) Use the answer to part (d) to get the value of h which minimizes the total cost.

Once you have the value of r that minimizes cost, plug that value into the formula for h you found in part (b). What do you get?

 

[niham]

The answer to part (c) will be a formula in the form c = (some function of r). This is what the problem is asking you to graph. If your calculator is anything like mine, it can only graph y as a function of x. But you can work around that by letting y = c and x = r. When you graph the function, what do you see? Approximately where is the minimum value? Is it a "nice" integer value?



Once you have the value of r that minimizes cost, plug that value into the formula for h you found in part (b). What do you get?[/QUOTE]

Yes it was not picometers it was supposed to be pi. So this is what I get for my equation, is this the same equation you get when you solve it?

(32)((2pir(800/pir^2) + (25)(2pir^2)
y= (51200/r) + 50pir^2
so I tried doing regression on ti I got r= -0.83
and so h= (800) /(pi)(-0.83)^2
so H = 369.6

 

[ksdhart2]

Your process is correct, and your answer is almost the same as what I got. I see just one small error. In your formula for part (b) you appear to be using the volume as 800 cubic meters, when it's meant to be 800 pi cubic meters, right? So the formula would be:

\(\displaystyle h=\dfrac{800 \cdot \pi}{\pi \cdot r^2}\)

You can thus cancel the pi terms. If you plug in that corrected value, you should get the right answer.

 

[niham]

Your process is correct, and your answer is almost the same as what I got. I see just one small error. In your formula for part (b) you appear to be using the volume as 800 cubic meters, when it's meant to be 800 pi cubic meters, right? So the formula would be:

\(\displaystyle h=\dfrac{800 \cdot \pi}{\pi \cdot r^2}\)

You can thus cancel the pi terms. If you plug in that corrected value, you should get the right answer.

Okay, so I did that but what Im really still not understanding which is that when you get the final equation which is 160849.54/r + 50pir^2
now if I put that in my calculator I can see a table already from my ti so if I just use like 10 values and find the r by linear regression this is what I then get.
so r = -0.86
h = (800)pi/pi(-0.749)^2
h= 1426

 

[niham]

The answer to part (c) will be a formula in the form c = (some function of r). This is what the problem is asking you to graph. If your calculator is anything like mine, it can only graph y as a function of x. But you can work around that by letting y = c and x = r. When you graph the function, what do you see? Approximately where is the minimum value? Is it a "nice" integer value?


so the minimum value is how I get r?

 

[ksdhart2]

So the minimum value is how I get r?

Sort of, but not quite. Let's carefully re-read the problem statement. "Use a graphing calculator to find the value of r which minimizes the total cost." The variable you want to minimize is c (y on the calculator). When you find the minimum value that c can possibly be, the corresponding r value is the answer to the question.