# AP Calculus

posted by .

An open box is to be made from cutting squares of side 's' from each corner of a piece of cardboard that is 35" by 40".

(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 -

length = 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.

## Similar Questions

1. ### Calculus

An open box is to be made from cutting squares of side "s" from each corner of a piece of cardboard 25" by 30". Write an expression for the volume, V, of the box in terms of s. -I have no idea where to start on this. I know V=lwh (length*width*height), …
2. ### PRE-CALCULUS

AN OPEN BOX IS FORMED BY CUTTING SQUARES OUT OF A PIECE OF CARDBOARD THAT IS 16 FT BY 19 FT AND FOLDING UP THE FLAPS. WHAT SIZE CORNER SQUARES SHOULD BE CUT TO YEILD A BOX THAT HAS A VOLUME OF 175 CUBIC FEET
3. ### calculus

Help!!! A rectangle piece of cardboard twice as long as wide is to be made into an open box by cutting 2 in. squares from each corner and bending up the sides. (a) Express the volume V of the box as a function of the width W of the …
4. ### math

a piece of cardboard is twice as it is wide. It is to be made into a box with an open top by cutting 2-in squares from each corner and folding up the sides. Let x represent the width (in inches) of the original piece of cardboard. …
5. ### Calculus

an open box is made by cutting out squares from the corners of a rectangular piece of cardboard and then turning up the sides. If the piece of cardboard is 12 cm by 24 cm, what are the dimensions of the box that has the largest volume …
6. ### Calculus

A box with an open top is to be made from a square piece of cardboard by cutting equal squares from the corners and turning up the sides. If the piece of cardboard measures 12 cm on the side, find the size of the squares that must …
7. ### calculus

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 14 in. long and 6 in. wide, find the dimensions of the box that …
8. ### calculus optimization problem

by cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. if the cardboard is 30 inches long and 14 inches wide find the dimensions of the box …
9. ### Engineering

A box with an open top is to be made from a square piece of cardboard by cutting equal squares from the corners and turning up the sides. If the piece of cardboard measures 12 in on the side, find the size of the squares that must …
10. ### Calculus 1

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 18 in. by 30 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume …

More Similar Questions