Here's the original question:
The diagram shows the net of a cylindrical container of radius r cm and height h. The full width of the metal sheet from which the container is made is 1 m, and the shaded area is waste. The surface area of the container is 1400 cm^2.
(i) Write down a pair of simultaneous equations for r and h.
(ii) Find the volume of the container, giving your answers in terms pi. (There are two possible answers.)
Thank you! However, we are unable to see "the diagram" or "the shaded area". Also, are we supposed to be
using the "waste", or is this stuff that doesn't count?
(i) The first answer I got was 6000pi which came from the first value of r=10. My working was:
. . .4r + h = 100
How did you obtain this equation? In particular, what is the source of the "100"?
. . .2 pi r h + 2 pi r^2 = 1400 pi
I will guess that this was derived as follows:
. . .The formula for the surface area SA of a cylinder
. . .with radius r and height h is:
. . . . .SA = 2(pi r^2) + (2 pi r)h
. . .We are given that SA = 1,400 pi square units, so:
. . . . .2 pi r^2 + 2 pi r h = 1,400 pi
If so, then I agree with your work and reasoning.
(ii) For this part, I started by using substitution:
. . .(2r - 20) (3r - 70) = 0
What did you substitute? Into where?
Assuming your first equation to be correct, I will guess that your work and reasoning was along these lines:
. . . . .4r + h = 100
. . .Then:
. . . . .h = 100 - 4r
. . .Plugging this in for "h" in the second equation, we get:
. . . . .2 pi r^2 + 2 pi r (100 - 4r) = 1,400 pi
. . .Dividing through by 2pi, we get:
. . . . .r^2 + r(100 - 4r) = 700
. . . . .r^2 + 100r - 4r^2 = 700
. . . . .100r - 3r^2 = 700
. . . . .0 = 3r^2 - 100r + 700
. . . . .0 = (3r - 70)(r - 10)
. . . . .r = 70/3 or r = 10
If so, then I agree with your work and reasoning.
From this there are two answers: r = 10 or r = 70/3. When r=10, h= 60; when r= 70/3, h= 100-(280/3).
Yes, but it would probably be better (easier for you, plus it's the answer the grader is expecting) to simplify the fractional form to 20/3.
This is what I got for the other value of r. I started from the answer that I got from the quadratic, 3r-70 =0. From that r= 70/3 putting this part into 4r+ h = 100 which gives the answer h= 100-(280/3) which is approx 6.67.
I'm sorry, but I'm not sure what you're doing here...? (By the way, pretty much
always you should use the "exact" form, rather than the decimal approximation, at least until the very end.)
The answer at the back says 6000pi or 98000pi/27 cm^2. I'm unsure how this answer came about and assumed it was from following the same process with the other value of r but I may be mistaken.
They plugged the exact forms of r and h (clearly, the forms they're wanting) into the volume formula.
So I have r and h and put it in the formula for a cylinder which gives 11375.95cm ^3, which then divided by pi is roughly 3621.1 pi.
What did you get, when you used the
exact form in the volume formula?
. . .V = pi r^2 h
. . . . .= pi (70/3)^2 (20/3)
. . . . .= pi (4900/9) (20/3)
. . . . .= pi [(4900 * 20) / (9 * 3)]
. . . . .= ....
...and so forth.