Math B/6.5
A rectangular packing box, similar to a shoebox without a lid, is three times as long as it is wide, and half as high as it is long. Each square inch of the bottom of the box costs $0.008 to produce, while each square inch of any side costs $0.003 to produce.
(a) Write a function that expresses the cost, c, of the box as a function of its width x.
(b) Using the function written in part (a), determine the dimensions of a box that would cost $0.69 to produce.
A rectangular packing box, similar to a shoebox without a lid, is three times as long as it is wide, and half as high as it is long. Each square inch of the bottom of the box costs $0.008 to produce, while each square inch of any side costs $0.003 to produce.
(a) Write a function that expresses the cost, c, of the box as a function of its width x.
(b) Using the function written in part (a), determine the dimensions of a box that would cost $0.69 to produce.
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Your facts are: l=3x, l=2h, 3x=2h, ba=0.008, sa=0.003
Your bottom area is: ba=lx and your side area is: sa=lh
so,
a) c=(3x^2)(0.008)+(9x/2)(0.003) per square inch
c=.024x^2+.0135x per square inch
b). 0.69=.024x^2+.0135x
Your bottom area is: ba=lx and your side area is: sa=lh
so,
a) c=(3x^2)(0.008)+(9x/2)(0.003) per square inch
c=.024x^2+.0135x per square inch
b). 0.69=.024x^2+.0135x
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x=3w h=1.5w
cost of bottom=0.008(3w)(w)
=0.024w²
cost of sides=2(x+w)(h)(0.003)
=2(4w)(1.5w)(0.003)
=0.036w²
a)
total cost=(0.024+0.036)w²
=0.06w²
b)
0.06w²=0.69
w²=0.69/0.06
w²=11.5
w= √11.5
w=3.391 in
h=5.087 in
length=10.173 in
cost of bottom=0.008(3w)(w)
=0.024w²
cost of sides=2(x+w)(h)(0.003)
=2(4w)(1.5w)(0.003)
=0.036w²
a)
total cost=(0.024+0.036)w²
=0.06w²
b)
0.06w²=0.69
w²=0.69/0.06
w²=11.5
w= √11.5
w=3.391 in
h=5.087 in
length=10.173 in
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Um... A shoebox?