Question

Water from a drinking fountain follows a parabolic path modeled by the equationh (t) = -49t² + 10t+ 0.1, where h representsthe height of the water above the basin and t tracks time in seconds.Part 1: Find the maximum height h of the water above the basin.Part 2: Find the time t it takes for water that exits the spout to land in the basin.Round each answer to the nearest hundredth.hfeetseconds

Answer 1

Given the equation:

`h(t)=-4.9t^2+10t+0.1`

Where h represents the height of water above the basin and t tracks time in seconds.

Let's answer the following questions:

• Part 1.

Find the maximum height h of the water above the basin.

To find the maximum height, take the derivative of h(t):

`\begin{gathered} h^(\prime)(t)=(d)/(dt)(-49t^2+10t+0.1) \\ \\ h^(\prime)(t)=98t+10 \\ \\ v(t)=h^(\prime)(t)=98t+10 \end{gathered}`

Where v(t) represents the velocity.

At maximum height, the velcoity is zero.

Now, plug in 0 for v(t) and solve for the time:

`\begin{gathered} 0=-98t+10 \\ \\ 98t=10 \\ \\ t=(10)/(98) \\ \\ t=(5)/(49) \end{gathered}`

Now plug in 5/49 for t in h(t) and solve:

`\begin{gathered} h((5)/(49))=-49((5)/(49))^2+10((5)/(49))+0.1 \\ \\ h((5)/(49))=-(25)/(49)+(50)/(49)+0.1 \\ \\ h((5)/(49))=0.610 \end{gathered}`

Therefore, the maximum height of the water above the basin is 0.61 feet.

• Part 2.

Find the time t it takes for water that exits the spout to land in the basin.

When the water lands the basin, the height, h(t) = 0.

To find the time, plug in 0 for h(t) and solve for t.

We have:

`\begin{gathered} 0=-49t^2+10t+0.1 \\ \\ 49t^2-10t-0.1=0 \end{gathered}`

SOlve uisng the quadratic formula:

Where:

a = 49

b = -10

c = -0.1

We have:

`\begin{gathered} t=(-b\pm√(b^2-4ac))/(2a) \\ \\ t=(-49\pm√(-10^2-4(49)(-0.1)))/(2(49)) \\ \\ t=(-49\pm√(100+19.6))/(98) \\ \\ t=(-49\pm√(119.6))/(98) \\ \\ t=(-49\pm10.94)/(98) \\ \\ t=(-49+10.94)/(98),(-49-10.94)/(98) \\ \\ t=0.214,\text{ -0.0096} \end{gathered}`

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

Therefore, the time is 0.214 seconds.

ANSWER:

Part 1: 0.610 feet

Part 2: 0.214 seconds