easy caluslus help, urgent

2010-03-11 9:16 pm

1.) There are currently 50 rabbits living on Lady Tottington’s Estate. Suppose
dR/dt = 0.04R where R represents the number of rabbits on the estate t months from now.
(a) Find a formula for R as a function of t.
(b) In how many months will there be 150 rabbits living on the estate?

2.)If a box must have a square end then what dimensions will give the box of
greatest volume which can be shipped via Priority Mail? The U. S. Postal Service will accept a box for shipment via Priority Mail only if the combined length and girth (distance around) is no more than 108 inches.

3.)A function f has second derivative f''(x) = (x − 1)^4(x − 2)^7(x^2 − 25). Determine
the x-value for each inflection point on the graph of f.

4.)A ladder 12 feet long rests against a vertical wall. If the bottom of the ladder
slides away from the wall at a rate of 0.5 feet per second, how quickly in radians per second is the angle between the ladder and the wall increasing when the bottom of the ladder is 5 feet from the wall?

回答 (1)

天助

2010-03-12 10:43 pm

✔ 最佳答案

Q1:
(a) dR/dt= 0.04R, R(0)=50
∫dR/R=∫ 0.04dt, then ln(R)= 0.04t+c,
put t=0, ln(50)=c, thus ln(R/50)=0.04t, R=50 exp(0.04t)
(b) R=150=50 exp(0.04t), then 0.04t= ln(3), t= 25ln(3) (months)

Q2:
What's the meaning of "square end" and "combined length"?

Q3:
f"(x)=0, then x=1, 2, 5, -5
f"(x) changes sign at x=-5, 2, 5
so the x-value of inflection points are x= -5, 2, 5
Note: x->1+, f"(x)>0 and x->1-, f"(x)>0

Q4:
Let the distance between the foot of the ladder and the wall be x(t) feet,
and the angle between the ladder and the wall be θ(t), then
x=12sin(θ) and dx/dt=0.5.
So, 12cos(θ)d/dt = 0.5, d/dt= 1/(24cosθ)
when x=5, sin(θ)=5/12, thus cos(θ)=√119/ 12
thus dθ/dt= 1/(2√119 ) (rad/sec)

2010-03-12 14:45:13 補充：
不好意思,來信提問,現在才回答
實在是抓不到Q2題意

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