Example 10.40 from Elementary Algebra, 10.4 Solve Applications Modeled by Quadratic Equations

Solve Applications Modeled by Quadratic Equations: example 10.40

Rene is setting up a holiday light display. He wants to make a ‘tree’ in the shape of two right triangles, as shown below, and has two 10-foot strings of lights to use for the sides. He will attach the lights to the top of a pole and to two stakes on the ground. He wants the height of the pole to be the same as the distance from the base of the pole to each stake. How tall should the pole be?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

The sun casts a shadow from a flag pole. The height of the flag pole is three times the length of its shadow. The distance between the end of the shadow and the top of the flag pole is 20 feet. Find the length of the shadow and the length of the flag pole. Round to the nearest tenth of a foot.

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Step 1. Read the problem. Draw a picture.
Step 2. Identify what we are looking for.		We are looking for the height of the pole.
Step 3. Name what we are looking for.		The distance from the base of the pole to either stake is the same as the height of the pole. Let $x =$ the height of the pole. $x =$ the distance from the pole to stake
Each side is a right triangle. We draw a picture of one of them.
Step 4. Translate into an equation. We can use the Pythagorean Theorem to solve for x.
Write the Pythagorean Theorem.		$a^{2} + b^{2} = c^{2}$
Step 5. Solve the equation. Substitute.		$x^{2} + x^{2} = 10^{2}$
Simplify.		$2 x^{2} = 100$
Divide by 2 to isolate the variable.		$\frac{2 x^{2}}{2} = \frac{100}{2}$
Simplify.		$x^{2} = 50$
Use the Square Root Property.		$x = \pm \sqrt{50}$
Simplify the radical.		$x = \pm 5 \sqrt{2}$
Rewrite to show two solutions.		$x = 5 \sqrt{2}$ $x = − 5 \sqrt{2}$
Approximate this number to the nearest tenth with a calculator.		$x \approx 7.1$
Step 6. Check the answer. Check on your own in the Pythagorean Theorem.
Step 7. Answer the question.		The pole should be about 7.1 feet tall.

Step 1. Read the problem. Draw a picture.
Step 2. Identify what we are looking for.		We are looking for the height of the pole.
Step 3. Name what we are looking for.		The distance from the base of the pole to either stake is the same as the height of the pole. Let $x =$ the height of the pole. $x =$ the distance from the pole to stake
Each side is a right triangle. We draw a picture of one of them.
Step 4. Translate into an equation. We can use the Pythagorean Theorem to solve for x.
Write the Pythagorean Theorem.		$a^{2} + b^{2} = c^{2}$
Step 5. Solve the equation. Substitute.		$x^{2} + x^{2} = 10^{2}$
Simplify.		$2 x^{2} = 100$
Divide by 2 to isolate the variable.		$\frac{2 x^{2}}{2} = \frac{100}{2}$
Simplify.		$x^{2} = 50$
Use the Square Root Property.		$x = \pm \sqrt{50}$
Simplify the radical.		$x = \pm 5 \sqrt{2}$
Rewrite to show two solutions.		$x = 5 \sqrt{2}$ $x = − 5 \sqrt{2}$
Approximate this number to the nearest tenth with a calculator.		$x \approx 7.1$
Step 6. Check the answer. Check on your own in the Pythagorean Theorem.
Step 7. Answer the question.		The pole should be about 7.1 feet tall.

Solve Applications Modeled by Quadratic Equations: example 10.40

Solution

Your turn

Solve Applications Modeled by Quadratic Equations: example 10.40

Solution

Your turn