+0  
 
0
51
3
avatar+81 

1.The consumers will demand 30 units when the price of a product is $53, and 47 units when the price is $33. Find the demand function (express thr price  in terms of the quantity ), assuming it is linear.

 

p=

 

2. A rectangular storage container with an open top is to have a volume of 80 cubic meters. The length of its base is twice the width. Material for the base costs 80 dollars per square meter. Material for the sides costs 9 dollars per square meter. Find the cost of materials for the cheapest such container. 

 

Minimum cost = 

 

3.The half-life of Radium-226 is 1590 years. If a sample contains 200 mg, how many mg will remain after 1000 years? 
 

 

Hint: Recall \( P = P_{0}e^{kt}v\)  where  \( P_{0}\) is the initial amount of such a substance, P is the amount at a given time t and k is the exponential rate of decay. What would it mean if \(P = .5 P_{0}\) ?

Sloan  Jun 7, 2018
edited by Sloan  Jun 7, 2018
edited by Sloan  Jun 7, 2018
 #1
avatar+86859 
+1

1.The consumers will demand 30 units when the price of a product is $53, and 47 units when the price is $33. Find the demand function (express thr price  in terms of the quantity ), assuming it is linear.

 

p=

 

We have the points  (30, 53)  and (47, 33)

 

We need to find the slope....so we have

 

[ 33 - 53]  / [ 47 - 30 ]  =  [ -20 ] / [ 17] =   -20/17

 

So we have

 

p  = (-20/17)(d - 30) + 53

 

p = (-20/17)d + 600/17 + 53

 

p = (-20/17)d + 1501/17

 

 

cool cool cool

CPhill  Jun 8, 2018
 #2
avatar+86859 
+1

2. A rectangular storage container with an open top is to have a volume of 80 cubic meters. The length of its base is twice the width. Material for the base costs 80 dollars per square meter. Material for the sides costs 9 dollars per square meter. Find the cost of materials for the cheapest such container. 

 

Minimum cost = 

 

The dimensions of the base  are   L * W   =  2W * W  = 2W^2

Ans the height  is H

 

The volume, V, can be expressed as

 

80  =  L * W * H

80  = 2W * W * H

80 = 2W^2 * H        solving for the height, we have

40/W^2  = H

 

So....the surface area,SA, can be expressed as

 

SA  = base area + area of the sides  = 

2W^2 + 2(40/W^2)(2W) + 2(40/W^2)(W)  =

2W^2 + 240/W

 

So....the cost, C, can be represented  as

 

C =

2W^2(80)  + (240/W)(9)  = 

160W^2  + 2160/W       take the derivative of this  and set to 0

 

 

C '  =  320W - 2160/W^2  = 0

320W^3 - 2160  =  0

W^3  =  2160/320

W^3  =  27/4

W  = 3/∛4 

 

And the minimum cost is given  by :

 

160(3/∛4)^2  + 2160 / (3/∛4 )   ≈  $1714.39

 

 

cool cool cool

CPhill  Jun 8, 2018
 #3
avatar+86859 
+1

3.The half-life of Radium-226 is 1590 years. If a sample contains 200 mg, how many mg will remain after 1000 years? 

 

We need to find k....[A  is some original amount]

 

.5A  = A * e^(k * 1590)   divide  through by A

.5 = e^(k * 1590)      take the Ln  of both sides

Ln (.5)  = Ln e^(k * 1590)  and we can write

Ln (.5)  = (k * 1590)    

k = Ln (.5) / 1590

 

So.....if the original amount is 200 mg, the amount remaining after 1000 years is

 

200* e^( Ln (.5) / 1590 * 1000)  ≈  129.3 mg

 

 

cool cool cool

CPhill  Jun 8, 2018

35 Online Users

avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.