1. A city official considers the demand for bus transportation (in number of bus

Question

1. A city official considers the demand for bus transportation (in number of bus riders) as a function of two variables: x = price of a bus ride in dollars, and y = price of parking in a downtown garage in dollars. She further hypothesizes that demand varies inversely with x and directly with y2. At present, bus fares are $.50, p is $6 per day, and there are 7200 riders using the system. arking a) Find fG,5). What does this number mean? b) Find and af 2, of c) Explain whatmeasures. Why is it negative? d) Explain what measures. Why is it positive? oy

Explanation / Answer

We can write this as :
f = k*y^2/x

Using x=0.5,y=6 and f = 7200, we get :
7200 = k*(6)^2 / 0.5
k = 100

So,
f = 100y^2/x

a)
f(0.5 , 5) :
f = 100*(5)^2/0.5 = 5000

So, 5000 riders

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b)
doe f / doe x is written as fx, the partial derivative of f with x
fx = -100y^2/x^2

fy = 100*(2y)/x
fy = 200y/x

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c)
fx measures the rate of change of f with x
keeping y constant.
It is negative because if the price of parking is fixed but yet you increase the bus fare, people are likely to not go in for it.
Also because f varies inversely with x, and thus fx is negative.

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d)
fy measures the rate of change of with y
keeping x constant.
f varies directly with y^2, as in with y increasing, f will increase too.
Thus fy was positive

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