A rectangular box has a base that is 4 times as long as it is wide. The sum of the height and the girth of the box is 200 feet. (a) Express the volume V of the box as a function of its width w. Determine the domain of V (w).

(a)

(b)

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"

"The sum of the height and the girth of the box is 200 feet"

The volume of the box is given by:

Using the L and H values from the equations above, we have:

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:

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.

problem:

f(x) = - 2/3x + 5; f(5/2)

substitute 5/2 in for x:

f(5/2) = -2/3(5/2) + 5

simplify the right side:

f(5/2) = -5/3 + 5

f(5/2) = 10/3

f(5/2) = 10/3

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