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The length of a 200 square foot rectangular vegetable garden is four feet less than twice the width. Find the length and width of the garden. Round to the nearest tenth of a foot.

The width of the garden is 11 feet and the length is 18 feet.

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A rectangular tablecloth has an area of 80 square feet. The width is 5 feet shorter than the length. What are the length and width of the tablecloth? Round to the nearest tenth of a foot.

The width of the tablecloth is 6.8 feet and the length is 11.8 feet.

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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.

Projectile motion

The height in feet, h , of an object shot upwards into the air with initial velocity, v 0 , after t seconds is given by the formula:

h = −16 t 2 + v 0 t

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 = −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 4.6 seconds, t 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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An arrow is shot from the ground into the air at an initial speed of 108 ft/sec. Use the formula h = −16 t 2 + v 0 t to determine when the arrow will be 180 feet from the ground. Round the nearest tenth of a second.

The arrow will reach 180 on its way up in 3 seconds, and on the way down in 3.8 seconds.

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A man throws a ball into the air with a velocity of 96 ft/sec. Use the formula h = −16 t 2 + v 0 t to determine when the height of the ball will be 48 feet. Round to the nearest tenth of a second.

The ball will reach 48 feet on its way up in .6 seconds and on the way down in 5.5 seconds.

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Access these online resources for additional instruction and practice with solving word problems using the quadratic equation:

Key concepts

  • Area of a Triangle For a triangle with base, b , and height, h , the area, A , is given by the formula: A = 1 2 b h
  • 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 = −16 t 2 + v 0 t
Practice Key Terms 2

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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