A soup can in the shape of a right circular cylinder is to be made from two materials. The material for the side of the can costs $0.015 per square inch and the material for the lids costs $ 0.027 per square inch. Suppose that we desire to construct a can that has a volume of 16 cubic inches. What dimensions minimize the cost of the can? a. Draw a picture of the can and label its dimensions with appropriate variables.
b. Use your variables to determine expressions for the volume, surface area, and cost of the can.
c. Determine the total cost function as a function of a single variable. What is the domain on which you should consider this function?
d. Find the absolute minimum cost and the dimensions that produce this value.
Answers
Answer:
a) file annex
b) V(c) = π*x²*y
A(x) = 2*π*x² + 32/x
C(x) = 0,1695*x² + 0,48 /x
Domain C(x) = {x/ x >0}
d) C(min) = 0,64 $
x = 1.123 in radius of base
y = 4,04 in height of the can
Step-by-step explanation:
See annex file
Lets:
call x = radius of the base of the cylinder and
y = the height of the cylinder
Then
Volume of the cylinder ⇒ V(c) = π*r²*h ⇒V(c) = π*x²*y
And y = V / ( π*x²) ⇒ V = 16 / ( π*x²)
Area of cylinder = lids area + lateral area
lids area = 2*π*x² ⇒ lateral area = 2*π*x*y
lateral area =2*π*x [16/(π*x²) ] ⇒ lateral area = 32/x
Then
A(x) = 2*π*x² + 32/x
Function cost C(x)
C(x) = 0.027 * 2*π*x² + 0.015 * (32/x)
C(x) = 0,1695*x² + 0,48 /x
Domain C(x) = {x/ x >0}
Now function cost:
C(x) = 0,1695*x² + 0,48 /x
Taking derivative:
C´(x) = 2*0,1695*x - 0.48/x² C´(x) = 0,339*x - 0.48/x²
C´(x) = 0 0.339*x³ - 0.48 = 0 x³ = 0.48/0.339 x³ = 1.42
x = 1.123 in
y = 16/πx² ⇒ y = 4,04 in
C(min) = 0,64 $
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1)
...
2)
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Answer:
$900 was the profit made for the month
Step-by-step explanation:
in this question, we are simply asked to calculate the amount of money a website made the owner if there is a certain number of subscriptions that month and a certain number of cancellations by old subscribers too.
From the question, we can see that 100 subscribers paid for subscription with 10 members canceling.
$10 is the subscription fee, the total amount made by the subscriptions is thus 10 * 100 = $1,000
in opposite light, there were 10 cancellations, the amount lost due to these would be 10 * 10 = $100
The profit made by the website would thus be ;
$1000 - $100 = $900
Answers
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