# easy caluslus help, urgent

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?

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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

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