求解决高中数学问题3.(a)Considerarationalfunctionp(x)withverticalasymptotesatx=3andx=−3.Thegraphofp(x)showsahorizontalasymptoteaty=4,andayinterceptatthepoint(0,−1).(i)Fromtheinformationpr

Question

求解决 高中数学问题 3. (a) Consider a rational function p(x) with
vertical asymptotes at x = 3 and x = −3. The graph of p(x) shows a horizontal
asymptote at y = 4, and a y intercept at the point (0, −1).
(i) From the information provided, write thefunction p(x) (4)
(ii) Using Autograph or otherwise, draw a
graph of the function p(x) including and 
labeling its vertical and horizontal
asymptotes as well as any intercepts. (3)
(b) A truck is on a 500 km journey. It is traveling
at a fairly constant speed on a reasonably straight straight and level road and
burns diesel fuel at a rate of g(x) litres per kilometre,
800+x2 where x is the average speed of the truck inkilometres per hour, and g(x) = 200x .
2980 + 3.725x2 (i) The cost of diesel is $1.49 perlitre. Show that C(x) = x
the fuel cost for this truck traveling at anaverage speed of x km/h.
where C is the fuel cost for this truck travelingat an average speed of x km/h.
(ii) Using Autograph, or otherwise, draw a
graph of C(x) with a y axis ranging from 0 to 500, and an x axis ranging from 0
to 80. Interpret and label each axis. (3)
(iii) From your graph, find the range of
average driving speeds, to the nearest whole number, that will keep the fuel
costs Cx) for the 500 km trip to $300 or less. (2)
(iv) From your graph, estimate to the nearest
whole number, the driving speed that will minimise the fuel cost for the trip.
Comment on its practicality, and anything that might invalidate this speed as
indicating minimum fuel usage. (3)
(v) Verify your answer in part (c)algebraically by solving the inequality, C(x) ≤ 300. (2)
4. A researcher is interested in the buying
behaviour (an indicator of demand) of a selected group of party pill users. In
a controlled experiment, a new legalised party pill is put on to the market.
Based on previous data, the researcher suggests the following demand model:
D(t) = 100 − 95 (3/5)t t is measured in the number of weeks it has been on the
market, and D(t) represents the percentage of the selected group wanting to buy
the party pill after t weeks (D(t) = 87, means 87% of the selected group). You
might find the graph of D(t) useful in helping you to answer the following
questions.
(i) Practically interpret what the modelsuggests about D(t) when t = 0. (1)
(ii) How many days after the party pill is
released does the model D(t) suggest that the 
demand for the party pill will
first exceed 50% of the selected group? (2)
(iii) After a short time though, the
researcher has to revise his model because real data received from the outlet
stores suggests that approximately 11 hours after the party pill went on the
market, 50% of the selected group had purchased it. He revises his model to
D(t) = 100 − 95(A)t. Find A and hence write the new model. (2)
(iv) Give a practical interpretation of theasymptotic behaviour for the function D(t) you found in (iii). 我正在翻译中 请稍等

Answer 1

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