Question

The height of a diver above the water’s surface can be modeled by the function h(t)= –16t^2+ 8t + 48. How long does it take the diver to hit the water? Solve by factoring

Answer 1

Given the function:

`h(t)=-16t^2+8t+48`

Where h(t) is the height of the diver above the surface of the water and t is the time.

Let's find how long it takes the diver to hit the water.

When the diver hits the water, the height h(t) = 0.

Now substitute 0 for h(t) and solve for the time t.

We have:

`0=-16t^2+8t+48`

Rearrange the equation:

`-16t^2+8t+48=0`

Solve for t.

Let's factor the expression by the left.

Factor 8 out of all terms:

`8(-2t^2+t+6)=0`

Now, factor by grouping.

Rewrite the middle term as a sum of two terms whose product is the product of the first term and the last term:

`\begin{gathered} 8(-2t^2+4t-3t+6)=0 \\ \end{gathered}`

Solving further:

`\begin{gathered} 8((-2t^2+4t)(-3t+6))=0 \\ \\ 8(2t(-t+2)+3(-t+2))=0 \\ \\ 8(2t+3)(-t+2)=0 \end{gathered}`

Hence, we have the factors:

`\begin{gathered} 2t+3=0 \\ -t+2=0 \end{gathered}`

Solve each factor for t:

`\begin{gathered} 2t+3=0 \\ \text{ Subtract 3 from both sides:} \\ 2t=-3 \\ \text{ Divide both sides by 2:} \\ (2t)/(2)=-(3)/(2) \\ t=-(3)/(2) \\ \\ \\ -t+2=0 \\ t=2 \end{gathered}`

Hence, we have the solutions:

t = -3/2

t = 2

The time cannot be negative, so let's take the positive value.

Therefore, the will take 2 seconds for the diver to hit the water.

ANSWER:

2 seconds.