Question

a scientist records the motion of a dolphin as it jumps from the water. the function ( h(t) = -16t^2 + 32t ) models the dolphin’s height in feet above the water after ( t ) seconds. for what time(s) ( t ) is the dolphin less than 12 feet above the water? you must show all of your work by algebraically solve the problem. you cannot simply graph and solve.

Explanation:

Step1: Set up inequality

$h(t) < 12$
$-16t^2 + 32t < 12$

Step2: Rearrange to standard form

Subtract 12 from both sides:
$-16t^2 + 32t - 12 < 0$
Multiply by -1 (reverse inequality):
$16t^2 - 32t + 12 > 0$
Divide by 4:
$4t^2 - 8t + 3 > 0$

Step3: Factor quadratic

Find factors of $4t^2-8t+3$:
$(2t - 1)(2t - 3) > 0$

Step4: Find critical points

Set factors to 0:
$2t - 1 = 0 \implies t = \frac{1}{2}$
$2t - 3 = 0 \implies t = \frac{3}{2}$

Step5: Test intervals

• For $t < \frac{1}{2}$: $(2t-1)(2t-3) > 0$ (true)
• For $\frac{1}{2} < t < \frac{3}{2}$: $(2t-1)(2t-3) < 0$ (false)
• For $t > \frac{3}{2}$: $(2t-1)(2t-3) > 0$ (true)

Also, $t \geq 0$ (time can't be negative). The dolphin lands when $h(t)=0$:
$-16t^2+32t=0 \implies t(16t-32)=0 \implies t=0$ or $t=2$. So $t \leq 2$.

Answer:

The dolphin is less than 12 feet above the water when $0 \leq t < \frac{1}{2}$ (0 to 0.5 seconds) and $\frac{3}{2} < t \leq 2$ (1.5 to 2 seconds).