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Solve each item completely.

I. Solve each item completely.
 
1. Shipping charges are often based on a formula that offers a lower charge per pound as the size of the shipment is increased. Suppose shipping charges are as follows: $2.20 per pound if the weight does not exceed 50 lb; $2.10 per pound if the weight is more than 50 lb but does not exceed 200 lb; $2.05 per pound if the weight is more than 200 lb.
             
         a. Find a mathematical model expressing the total cost of a shipment as a function of its weight.
 
 
2. Due to the curvature of the earth, the maximum distance D that you can see from the top of a tall building or from an airplane at height h is given by the function
                                             D(h)= √(2rh+h^2 )
where r =3960 mi is the radius of the earth and D and h are measured in miles.
 
         a.      How far can you see from the observation deck of Toronto’s CN Tower, 1135 ft above the ground?
         b.      Commercial aircraft fly at an altitude of about 7 mi. How far can the pilot see?
 
3. A person’s approximate brain weight is directly proportional to his or her body weight, and a person weighing 150 lb has an approximate brain weight of 4lb.
         a. Find a mathematical model expressing the approximate brain weight of a person as a function of the                 person’s body weight.
          b. Find the approximate brain weight of a person whose body weight is 176 lb.
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1 Answer

For problem 1, there are three scenarios possible:
 
1) 0 < W <= 50  then cost per pound = $2.20
2) 50 < W <= 200 then cost per pound = $2.10
3) 200 < W then cost per pound = $2.05
 
In general the cost of shipping is:
 
Cost = W * (Cost Per Pound)
 
If we have a W = 40 LBS then
 Cost = 40 * $2.20 = $80.80
 
If we have a W = 60 LBS then
 Cost = 60 * $2.10 = $120.60
 
If we have a W = 300 then
 Cost = 300 * $2.05 = $603.00
 
Hopefully that makes sense.....now, to write that a a function i would do something like this:
 
f(w) = 2.20 * w , 0    < w <= 50
          2.10 * w, 50   < w <= 200
          2.05 * w, 200 < w
 
 
 
For problem 2:
 
D(h)= √(2rh+h^2 )
 
First, we need to get things into miles since they tell us that the equation expects miles
so 1135 ft = 1135/5280 miles = 0.215 mi
 
D(h) = ( 2(3960)(.215) + .215^2 ) ^0.5 = 41.23 miles
 
D(h) = ( 2(3960)(7) + 7^2 )^0.5 = 327.06 miles
 
 
Problem 3:
Brain (B) is proportional to Body Weight (W) or B/W
 
B = 4 when W = 150 so we know that B/W = 4/150
 
Part A)
    B = (B/W) * W
 
    check it with given data:    B = (4/150) * 150 = 4.....correct.
 
Part B)
   B = (4/150) * 176 = 4.7
 
 
 
 

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