answer in detail please, thank you If you were to grow plants in a rectangular-s

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answer in detail please, thank you

If you were to grow plants in a rectangular-shaped garden with a perimeter of 10 ft, what is the maximum area you can achieve? Assume that the function H(t) gives the height in ft of a stone thrown vertically upward at a velocity of 20 ft/s. How high does the stone go? H(t) = -4.9t2 + 20t + 2 Back at work as a clown, you are pumping gas into a spherical balloon so that its volume increases at a rate of 2 in 3/s. What is the rate of change of its radius when it is 6 inches? Assume that ABC Airlines has a policy that states the baggage must be box-shaped and its sum of height, width and length must not exceed 108 inches. 1) What dimensions (length, width, height) would give the maximum volume and 2) what is the maximum volume? Given that a company's net sales in billion is given by S(t) where t is the number of years since 2000. What were the 1) net sales and 2) rate of change in sales in year 2010?

Explanation / Answer

a)perimeter =2(a+b)=10 so a+b=5ft

area=ab=a(5-a)

to maximize area d(ab)/da=0=>5-2a=0=>a=2.5 ft

so b=2.5 therefore area =2.5*2.5=6.25 ft^2

b initial velocity v0=20

at max height v=0=>dh/dt=0

-9.8t+20=0=>t=2.04s

so h at t=2.04 s ..put t=2.04s in H(t)

c) v=4*pi*r^3/3

=>dv/dt=4*pi*r^2*dr/dt

given dv/dt=2 ,, r=6=>dr/dt=2/(4*36*3.14)=0.0044

d) a+b+c=108

by arithmetic geometric mean theorem

(a+b+c)/3>=cube root (abc)=>abc<=36^3 equality holds for a=b=c.

so maximum volume =36*36*36

e) 1) for net sales put x=10 in s(t)

2) for rate of change dS/dt=-0.75x^2+0.46x+10=-60.4

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