Question

The polynomial -x^3-x^2+12x represents the volume of a rectangular aqautic tank in cubic feet. the length of the tank is (x+4).

Answer 1

Answer:

a. It should look for 3 linear factors.

b. There are 3 dimensions.

c. x = 1.7

d. Maximum value is 12.597

Step-by-step explanation:

a.

$-x^{3} - x^{2} + 12 \timesx$ = $-x^{3} - 4 \timesx^{2} + 3 \timesx^{2} +12 \timesx$ = $-(x + 4) \timesx^{2} + 3 \timesx \times(x + 4)$ = $(x + 4) \timesx \times (x + 3)$

Hence the factorized form of the volume of the poly is $(x + 4) \timesx \times (x + 3)$

3 linear factors should needed to be look for.

b.

The 3 dimensions are (x + 4), x, (x + 3).

c.

Let, f(x) = $-x^{3} - x^{2} + 12 \timesx$

Now, $\frac{d f(x)}{dx}$ = $-3 \timesx^{2} -2 \timesx + 12$

Let $f1(x)$ = $-3 \timesx^{2} -2 \timesx + 12$ and $f2(x)$ = $\frac{d f1(x)}{dx}$ = $-3 \timesx -2$

The value will be maximum, if f1(x) = 0 and f2(x) < 0.

$f1(x) = 0$\\ $3 \timesx^{2} + 2 \timesx -12 = 0$\\ $x = \frac{- 2 + \sqrt{2^{2} + 4 \times3 \times12} }{2 \times3} , \frac{- 2 - \sqrt{2^{2} + 4 \times3 \times12} }{2 \times3}$

x = 1.69 ≅ 1.7 or x = -2.36 ≅ -2.4

Now f2(1.7) < 0.

At x = 1.7, the volume will be maximum.

d.

The maximum value is f(1.7) = 12.597(approximately)