Question

the function $v(h) = h(14 - 2h)(16 - 2h)$ represents the volume, in cubic inches, of a rectangular box.
it increases then decreases then increases
the manufacturer wants to make a box with a height no greater than 9 inches that has a maximum volume.
what height should the manufacturer use? what is the volume of the box? round your answers to the nearest hundredth.

enter the answer in each space provided. use numbers instead of words

height inches
volume cubic inches

Explanation:

Step1: Expand the volume function

First, expand $V(h) = h(14-2h)(16-2h)$:

1. Multiply $(14-2h)(16-2h)$:

$$(14)(16) + 14(-2h) -2h(16) + (-2h)(-2h) = 224 - 28h -32h +4h^2 = 4h^2 -60h +224$$

1. Multiply by $h$:

$$V(h) = h(4h^2 -60h +224) = 4h^3 -60h^2 +224h$$

Step2: Find derivative of $V(h)$

Take the first derivative to find critical points:
$$V'(h) = \frac{d}{dh}(4h^3 -60h^2 +224h) = 12h^2 -120h +224$$

Step3: Solve $V'(h)=0$ for $h$

Use quadratic formula for $12h^2 -120h +224=0$, where $a=12$, $b=-120$, $c=224$:
$$h = \frac{-b \pm \sqrt{b^2-4ac}}{2a} = \frac{120 \pm \sqrt{(-120)^2 -4(12)(224)}}{2(12)}$$
Calculate discriminant:
$$\sqrt{14400 - 10752} = \sqrt{3648} \approx 60.3987$$
Solve for $h$:
$$h_1 = \frac{120 + 60.3987}{24} \approx 7.5166$$
$$h_2 = \frac{120 - 60.3987}{24} \approx 2.4834$$
We also consider the domain: $h>0$, $14-2h>0 \Rightarrow h<7$, $16-2h>0 \Rightarrow h<8$, and $h \leq9$. So valid critical points are $h\approx2.48$ and $h\approx7.52$ (but $h\approx7.52$ violates $h<7$, so discard it). We also check endpoints $h=0$ (volume=0) and $h=7$ (volume=$7(14-14)(16-14)=0$).

Step4: Test critical point for maximum

Second derivative test:
$$V''(h) = 24h -120$$
At $h\approx2.4834$:
$$V''(2.4834) = 24(2.4834)-120 \approx 59.6016 -120 = -60.3984 <0$$
This confirms a local maximum at $h\approx2.48$.

Step5: Calculate maximum volume

Substitute $h\approx2.48$ into $V(h)$:
$$V(2.48) = 2.48(14-2(2.48))(16-2(2.48))$$
Calculate each term:
$$14-4.96=9.04, \quad 16-4.96=11.04$$
$$V(2.48) = 2.48 \times 9.04 \times 11.04 \approx 2.48 \times 99.7996 \approx 247.50$$

Step6: Analyze volume trend

As $h$ increases from 0 to 2.48, volume increases; as $h$ increases from 2.48 to 7, volume decreases; for $h>7$, volume is negative (invalid, since volume can't be negative). So the trend is: increases, then decreases, then (invalid, so we ignore the last increase as it's not a real volume).

Answer:

It increases, then decreases, then (invalid, so the valid trend is) increases, then decreases
Height: 2.48 inches
Volume: 247.50 cubic inches