Modeling Data
The circumference C (in inches) of a vase is measured at three-inch intervals starting at its base. The measurements are shown in the table, where y is the vertical distance in inches from the base.
y
0
3
6
9
12
15
18
C
50
65.5
70
66
58
51
48
(a) Use the data to approximate the volume of the vase by summing the volumes of approximating disks.
(b) Use the data to approximate the outside surface area (excluding the base) of the vase by summing the outside surface areas of approximating frustums of right circular cones.
(c) Use the regression capabilities of a graphing utility to find a cubic model for the points (y, r), where r = C / ( 2 π ) . Use the graphing utility to plot the points and graph the model.
(d) Use the model in part (c) and the integration capabilities of a graphing utility to approximate the volume and outside surface area of the vase. Compare the results with your answers in parts (a) and (b).
Modeling Data
The circumference C (in inches) of a vase is measured at three-inch intervals starting at its base. The measurements are shown in the table, where y is the vertical distance in inches from the base.
|
y |
0 |
3 |
6 |
9 |
12 |
15 |
18 |
|
C |
50 |
65.5 |
70 |
66 |
58 |
51 |
48 |
(a) Use the data to approximate the volume of the vase by summing the volumes of approximating disks.
(b) Use the data to approximate the outside surface area (excluding the base) of the vase by summing the outside surface areas of approximating frustums of right circular cones.
(c) Use the regression capabilities of a graphing utility to find a cubic model for the points (y, r), where r = C / ( 2 π ) . Use the graphing utility to plot the points and graph the model.
(d) Use the model in part (c) and the integration capabilities of a graphing utility to approximate the volume and outside surface area of the vase. Compare the results with your answers in parts (a) and (b).
