a pie dish has base diameter 8 inches and top diameter 10 inches and height 2 inches.
what is the volume of the disk?
do I do it so cross sections are squares?
help!!
Thanks!
I would visualize the pie dish as a cut-off part of a cone, which has been sliced horizontal to the base, 2 inches up from the base
let the distance to the vertex of the imaginary cone be x inches.
then by ratios:
5/(x+2) = 4/x
x = 8
volume of cone with height of 10 and diameter of 10 = 1/3(pi)(5^2)(10) = (250/3)pi
volume of the cone which is cut off
= 1/3pi(4^2)(8) = (128/3)pi
so the volume of the pie dish is
(250/3)pi - (128/3)pi = 122/3pi
= appr. 127.76 cu. in.
Let AreaBase, AreaTop be the area of the bottome and top. Then the area of the center (1/2 way up) is AreaMiddle
A very old approximating formula for volume of anything, even irregular objects, is
Volume= 1/6(areabottom+ 4AreaMiddle+ AreaTop)* height.
For instance, the area of a sphere: Areabottom, areatop is zero, area middle is PI r^2
Then the approximating formula is
Volume= 1/6(4PIr^2)2r= 4/3 PI r^3
That ought to put a kink in the profs text.
Now the calculus way:
Do it in cross sections. Relate r to height (r=4+h/2) so dr= dh/2
Then check it with that old approximating formula schoolboys memorized several hundred years ago.
Volume= INT PI(4+h/2)^2 dh
=INT PI (16+4h+h^2/4) dh
= PI(16h + 2h^2 + h^3/12 ) eval at the limits, or
Volume= PI(32+8+8/12)= 40.75PI by calculus, check my thinking.
Now the old approximating formula:
Volume= 1/6(PI)(16 + 4*20.25+ 25)2
1/6 PI (61.25)2=40.67 PI
That old formula is handy.
Did you mean using shell method? Using shells is quite complicated for this problem :/.
Vertical
Height = x
Length = 2yPi = 2(-2x+10)Pi
Width = dx
Take the integral of 2x(-2x+10)Pidx from 4 to 5 and you get 26/3Pi. Then take the integral of 2(2xPi)dx from 0 to 4 and you get 32Pi. Add them together and you get 122/3Pi. Which is the same as the other answers.
Horizontal(took me well over 30 minutes -.- Didn't know I was so weak at shells..)
Height = 2-x
Length = 2Pi(.5x+4)
Width = dx
Integrate from 0 to 2 and add the volume of the inside cube which is 32Pi and you'll get the same answer.
To find the volume of the pie dish, you need to consider the shape of the cross sections. In this case, you can approximate the cross sections as frustums of cones or truncated cones.
To calculate the volume of each frustum, you can use the formula for the volume of a frustum of a cone: V = 1/3 * π * h * (R^2 + r^2 + Rr), where h is the vertical height of the frustum, R and r are the radii of the larger and smaller bases, respectively.
In this case, since the pie dish is a disk with a constant height, the cross sections are all identical frustums, each with a height of 2 inches. The larger base radius (R) would be equal to half the top diameter (10 inches/2 = 5 inches), and the smaller base radius (r) would be equal to half the base diameter (8 inches/2 = 4 inches).
So, you can calculate the volume of the pie dish by summing up the volumes of all the frustums. However, since the cross sections are squares, this formula doesn't directly apply. Instead, you can approximate the shape of each cross section as a rectangular prism and calculate the volume as the area of each cross section multiplied by the height.
To approximate the cross section shape as a rectangle, you can use the fact that the circumference of the larger base is approximately equal to the perimeter of the rectangular cross section. The circumference of the larger base can be calculated as 2 * π * R, where R is the larger base radius (5 inches).
The formula for the perimeter of a rectangle is 2 * (length + width). In this case, the "width" of the rectangle would be equal to the height of the pie dish (2 inches), and the "length" would be equal to the unknown length of the cross section.
Equating the circumference of the larger base to the perimeter of the rectangle, you can get the formula: 2 * length + 2 * 2 = 2 * π * R.
Simplifying the equation, you can solve for "length": length = π * R - 2.
Once you have the length of the cross section, you can calculate the area of the cross section as length * width, which would be (π * R - 2) * 2.
Finally, to find the volume of the pie dish, you would need to sum up the volumes of all the cross sections (each cross section being a rectangular prism). You can do this by multiplying the area of each cross section by the height of the pie dish (2 inches) and summing up these volumes.
I hope this explanation helps! Let me know if you have any further questions.