Answer:
The approximate value of w that maximize the volume
V = -w³ + w² + 20w
is w = 3 units
Explanation:
Volume of rectangular prism is given by the formula:
V = Bh
B is the base, and
B = lw
Where l is the length of the rectangle, and w is the width.
B = w(w + 4)
= w² + 4w
V = Bh
= (w² + 4w)(5 - w)
= 5w² - w³ + 20w - 4w²
V = -w³ + w² + 20w
To obtain the approximate value of w that maximizes the volume, let us solve:
-w³ + w² + 20w = 0
w(-w² + w + 20) = 0
w = 0
Or
-w² + w + 20 = 0
Using the quadratic formula,
w = [-b±√(b² - 4ac)]/2a
Where a = -1, b = 1, and c = 20.
w = {-1 ±√[(1 - 4(1)(20)]}/2(-1)
= (-1 ±√25)/(-2)
= 1/2 ± -5/2
w = 1/2 + 5/2
= -4/2 = -2
w = 1/2 - 5/2
= 6/2 = 3
Now we have the following values for w.
w = 0 or -2 or 3
The maximum of these is 3, and it is what we are looking for.