搜搜考试网美国微积分问题（懂英语的来）
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美国微积分问题（懂英语的来）
The cost of fuel to propel a boat through the water（in dollars per hour)is proportional to the cube of the speed.A certain ferry boat uses $100 worth of fuel per hour when crusing at 10 miles per hour.Apart from fuel,the cost of running this ferry(labor,maintenance,and so on) is $675 per hour.AT what speed should it travel so as to minimize the cost per mile traveled?
The cost of fuel to propel a boat through the water（in dollars per hour)is proportional to the cube of the speed.A certain ferry boat uses $100 worth of fuel per hour when crusing at 10 miles per hour.Apart from fuel,the cost of running this ferry(labor,maintenance,and so on) is $675 per hour.AT what speed should it travel so as to minimize the cost per mile traveled?
Solution:
Let the cost of fuel be x
x ∝ v³
Let x = kv³,where k is the proportionality.
Sub v = 10 mi/hr,x = 100 into x = kv³,
we have k = 100/1000 = 1/10
So,x = v³/10
Let the total cost per mile be y.
The time taken for one mile = 1/v.
y = (1/v)×v³/10 + (1/v)×675
= v²/10 + 675/v
dy/dv = v/5 - 675/v²
Let dy/dv = 0
so v³ = 675×5 = 3375,
v = 15 (mi/hr)
d²y/dv² = 1/5 + 1350/v > 0
So,when v = 15 (mi/hr)
the minimized cost per mile is
ymin = 15²/10 + 675/15 = 67.5($)
Let the cost of fuel be x
x ∝ v³
Let x = kv³,where k is the proportionality.
Sub v = 10 mi/hr,x = 100 into x = kv³,
we have k = 100/1000 = 1/10
So,x = v³/10
Let the total cost per mile be y.
The time taken for one mile = 1/v.
y = (1/v)×v³/10 + (1/v)×675
= v²/10 + 675/v
dy/dv = v/5 - 675/v²
Let dy/dv = 0
so v³ = 675×5 = 3375,
v = 15 (mi/hr)
d²y/dv² = 1/5 + 1350/v > 0
So,when v = 15 (mi/hr)
the minimized cost per mile is
ymin = 15²/10 + 675/15 = 67.5($)