Respuesta :
Answer:
(a) [tex]V = (-8W^3 + 800W^2)/3[/tex]
(b) [tex]W > 100[/tex]
Step-by-step explanation:
Let's call the length of the box L, the width W and the height H. Then, we can write the following equations:
"A rectangular box has a base that is 4 times as long as it is wide"
[tex]L = 4W[/tex]
"The sum of the height and the girth of the box is 200 feet"
[tex]H + (2W + 2H) = 200[/tex]
[tex]2W + 3H = 200 \rightarrow H = (200 - 2W)/3[/tex]
The volume of the box is given by:
[tex]V = L * W * H[/tex]
Using the L and H values from the equations above, we have:
[tex]V = 4W * W * (200 - 2W)/3[/tex]
[tex]V = (-8W^3 + 800W^2)/3[/tex]
The domain of V(W) is all positive values of W that gives a positive value for the volume (because a negative value for the volume or for the width doesn't make sense).
So to find where V(W) > 0, let's find first when V(W) = 0:
[tex](-8W^3 + 800W^2)/3 = 0[/tex]
[tex]-8W^3 +800W^2 = 0[/tex]
[tex]W^3 -100W^2 = 0[/tex]
[tex]W^2(W -100) = 0[/tex]
The volume is zero when W = 0 or W = 100.
For positive values of W ≤ 100, the term W^2 is positive, but the term (W - 100) is negative, then we would have a negative volume.
For positive values of W > 100, both terms W^2 and (W - 100) would be positive, giving a positive volume.
So the domain of V(W) is W > 100.
