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August 1, 2014

August 1, 2014

Posted by **Casey** on Wednesday, July 17, 2013 at 7:49pm.

(a) Write an expression for the volume, 'V', of the box in terms of 's'.

(b) Draw a graph of V(s).

(c) State the domain and range of V(s).

(d) Find the value of 's' that will give the maximum volume. What is the maximum volume?

(e) What realistic value(s) of 's' will generate a volume of 1225 cubic units?

- AP Calculus -
**Reiny**, Wednesday, July 17, 2013 at 11:09pmlength = 40-2s

width = 35-2s

height = s

a) V = s(40-2s)(35-2s)

= ..

= 1400s - 150s^2 + 4s^3

b) standard shape of a cubic, with x-intercepts of

0 , 20, and 17.5

c) make special notice of where V is above and below the x-axis

d) dV/ds = 1400 - 300s + 12s^2

= 0 for a max of V

divide by 4 ...

3s^2 - 75s + 350 = 0

s = (75 ± √1425)/6

= appr 18.8 or appr 6.2

clearly the 18.8 would produce a negative width, so we reject that

accept s = 6.2 to yield a max volume of appr 3867 cubic inches

to get a volume of 1225 we set

4s^3 - 150s^2 + 1400s = 1225

4s^3 - 150s^2 + 1400s - 1225 = 0 gives a solution of

s = .974 , 13.9 and 22.6

by Wolfram:

http://www.wolframalpha.com/input/?i=4s%5E3+-+150s%5E2+%2B+1400s+-+1225%3D0

both .974 and 13.9 yield our needed result, while 22.6 would produce a negative length and width, thus we must reject it.

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