10.4 Solve applications modeled by quadratic equations  (Page 4/7)

The height of a projectile shot upwards is modeled by a quadratic equation. The initial velocity, $v_0$ , propels the object up until gravity causes the object to fall back down.

We can use the formula for projectile motion to find how many seconds it will take for a firework to reach a specific height.

A firework is shot upwards with initial velocity 130 feet per second. How many seconds will it take to reach a height of 260 feet? Round to the nearest tenth of a second.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.	We are looking for the number of seconds, which is time.
Step 3. Name what we are looking for.	Let $t=$ the number of seconds.
Step 4. Translate into an equation.	Use the formula.
		$h=\mathrm{-16}{t}^2+v_{0}t$
Step 5. Solve the equation. We know the velocity $v_0$ is 130 feet per second.
The height is 260 feet. Substitute the values.
This is a quadratic equation, rewrite it in standard form.
Solve the equation using the Quadratic Formula.
Identify the a, b, c values.
Write the Quadratic Formula.
Then substitute in the values of a, b, c .
Simplify.
Rewrite to show two solutions.
Approximate the answers with a calculator.	$t\approx 4.6$ seconds, $t\approx 3.6$
Step 6. Check the answer. The check is left to you.
Step 7. Answer the question.	The firework will go up and then fall back down. As the firework goes up, it will reach 260 feet after approximately 3.6 seconds. It will also pass that height on the way down at 4.6 seconds.

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Key concepts

• Area of a Triangle For a triangle with base, $b$ , and height, $h$ , the area, $A$ , is given by the formula: $A=\frac{1}{2}bh$
  
• Pythagorean Theorem In any right triangle, where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypothenuse, $a^2+b^2=c^2$
  
• Projectile motion The height in feet, $h$ , of an object shot upwards into the air with initial velocity, $v_0$ , after $t$ seconds can be modeled by the formula:
  
  $h=\mathrm{-16}{t}^2+v_{0}t$