سؤال في تطبيقات التكامل والتفاضل

[memee-a]

A circular cylindrical container open at the top has capacity . If the cost of the material used for the bottom is twice that of the material used for curved part , find the dimensions of the container so that the cost is minimal .

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A heated rod with temperature 200 C is left to cool down in a room with temperature 30 C . Suppose the temperature of the rod at time t is modeled by where A,B and K are constants and t is measured in minutes.
If the temperature of the rod falls to 100 C after 10 minutes , compute A,B and K
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[memee-a]

A circular cylindrical container open at the top has capacity file:///C:/DOCUME%7E1/a/LOCALS%7E1/Temp/msohtml1/01/clip_image002.gif. If the cost of the material used for the bottom is twice that of the material used for curved part , find the dimensions of the container so that the cost is minimal .

Area of bottom = π r^2
Area of the current part = 2 π r h
Capacity = 16 π
To find minimum cost let us find minimum area
C = 2 π r h + 2 π r^2
But (v) = π r^2 h = 16 π à h = 16 π / π r^2 = 16 / r^2 (1)
So c = 2 π r 16 / r^2 + 2 π r^2
C = 100.48 / r + 2 π r^2
C` = -100.48 / r^2 + 4 π r
C` = 0 à - 100.48 + 4 π r = 0
-100.48 + 4 π r^3 = 0
r^3 = 100.48/ 4 π = 8
r = 2
minimum
c`` = 200.96/r^3 + 4 π
c`` (2) = 37.68
Substituted
h = 16 / r^2
h = 16/ 4
h = 4
The dimension of the container give minimum cost is
r = 2 , h = 4

[memee-a]

A heated rod with temperature 200 C is left to cool down in a room with temperature 30 C . Suppose the temperature of the rod at time t is modeled by file:///C:/DOCUME%7E1/a/LOCALS%7E1/Temp/msohtml1/01/clip_image002.gif where A,B and K are constants and t is measured in minutes.
If the temperature of the rod falls to 100 C after 10 minutes , compute A,B and K
(2 points each).

T (t) = A + B e^(-kt)
T (0) = 200
T (10) = 100
A = the time of decay = 30°c
T (0) = 200 = A + B e^(-kt) à A + B = 200
B = 200 – A
B = 200 – 30
B = 170
T (t) = 30 + 170 e ^(-kt)
To find k
T (10) = 100
100 = 30 + 170 e ^(-10k)
70 = 170 e^(-10k)
- 10k = ln(70/170)
K = -0.88/-10
K = 0.088
T (t) = 30 + 170 e^(0.088t)

[memee-a]

this is my answer i hope it’s right