A swimmer dives into a lake from a cliff and swims to the surface in a parabolic
path. The path of the swimmer can be modeled by the equation
, where is the height in feet, and is the time in
seconds since the swimmer left the top of the cliff.
Complete the sentence.
The diver will be underwater for a total of

Ask by Cervantes Olson. in the United States

Mar 25,2025

Question

A swimmer dives into a lake from a cliff and swims to the surface in a parabolic
path. The path of the swimmer can be modeled by the equation
, where is the height in feet, and is the time in
seconds since the swimmer left the top of the cliff.
Complete the sentence.
The diver will be underwater for a total of

Ask by Cervantes Olson. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

To determine how long the diver will be underwater, we need to find the time $$ t $$ when the swimmer is at a height of $$ h(t) = 0 $$ (the surface of the water). This involves solving the equation:

$$
-t^{2} - 10t + 9 = 0
$$

Let's solve this quadratic equation for $$ t $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$-t^{2}-10t+9=0$$
- step1: Multiply both sides:
  $$t^{2}+10t-9=0$$
- step2: Solve using the quadratic formula:
  $$t=\frac{-10\pm \sqrt{10^{2}-4\left(-9\right)}}{2}$$
- step3: Simplify the expression:
  $$t=\frac{-10\pm \sqrt{136}}{2}$$
- step4: Simplify the expression:
  $$t=\frac{-10\pm 2\sqrt{34}}{2}$$
- step5: Separate into possible cases:
  $$\begin{align}&t=\frac{-10+2\sqrt{34}}{2}\&t=\frac{-10-2\sqrt{34}}{2}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&t=-5+\sqrt{34}\&t=\frac{-10-2\sqrt{34}}{2}\end{align}$$
- step7: Simplify the expression:
  $$\begin{align}&t=-5+\sqrt{34}\&t=-5-\sqrt{34}\end{align}$$
- step8: Rewrite:
  $$t_{1}=-5-\sqrt{34},t_{2}=-5+\sqrt{34}$$
The solutions to the equation $$ -t^{2} - 10t + 9 = 0 $$ are:

$$
t_{1} = -5 - \sqrt{34}
$$
$$
t_{2} = -5 + \sqrt{34}
$$

Since time cannot be negative, we will only consider $$ t_{2} $$ for the time when the swimmer reaches the surface.

Next, we need to calculate $$ t_{2} $$ to find the time the swimmer is underwater. Let's compute $$ t_{2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$-5+\sqrt{34}$$
The value of $$ t_{2} $$ is approximately $$ 0.831 $$ seconds.

To find the total time the diver is underwater, we need to consider the time from when the swimmer dives in until they reach the surface. Since the swimmer starts at the top of the cliff (at $$ t = 0 $$) and reaches the surface at $$ t \approx 0.831 $$ seconds, the total time underwater is approximately $$ 0.831 $$ seconds.

Thus, we can complete the sentence:

The diver will be underwater for a total of approximately $$ 0.831 $$ seconds.
The diver will be underwater for approximately 0.831 seconds.

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