Problem 37 - Precalculus Student Solutions Manual 5th

Question

A nature conservancy group decides to construct a raised wooden walkway through a wetland area. To enclose the most interesting part of the wetlands, the walkway will have the shape of a right triangle with one leg 700 yd longer than the other and the hypotenuse 100 yd longer than the longer leg. Find the total length of the walkway.

Step-by-Step Solution

Answer

The total length of the walkway is 3000 yards.

Step 1 - Define Variables

Let the shorter leg of the triangle be denoted as \(x\) yards. Then, the longer leg will be \(x + 700\) yards, and the hypotenuse will be \(x + 700 + 100 = x + 800\) yards.

Step 2 - Set Up the Equation

Using the Pythagorean theorem, the equation for this right triangle would be: \[ x^2 + (x + 700)^2 = (x + 800)^2 \]

Step 3 - Expand and Simplify

Expand the squares and combine like terms: \[ x^2 + (x^2 + 1400x + 490000) = x^2 + 1600x + 640000 \] Combine like terms: \[ 2x^2 + 1400x + 490000 = x^2 + 1600x + 640000 \]

Step 4 - Solve the Equation

Subtract \(x^2\) and rearrange the equation: \[ x^2 + 1400x + 490000 = 1600x + 640000 \] \[ x^2 - 200x - 150000 = 0 \]

Step 5 - Use Quadratic Formula

Solve the quadratic equation \(x^2 - 200x - 150000 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -200\), and \(c = -150000\). \[ x = \frac{200 \pm \sqrt{200^2 + 600000}}{2} \] \[ x = \frac{200 \pm \sqrt{640000}}{2} \] \[ x = \frac{200 \pm 800}{2} \]

Step 6 - Calculate the Solutions

The solutions are: \[ x = \frac{200 + 800}{2} = 500 \] \[ x = \frac{200 - 800}{2} = -300 \] Since length cannot be negative, \(x = 500\) yards.

Step 7 - Find the Lengths of the Triangle

The lengths of the legs are: Shorter leg: \(x = 500\) yards Longer leg: \(500 + 700 = 1200\) yards Hypotenuse: \(500 + 800 = 1300\) yards.

Step 8 - Calculate the Total Length of the Walkway

To find the total length of the walkway, add the lengths of the three sides: \[ 500 + 1200 + 1300 = 3000 \text{ yards} \]

Key Concepts

right triangle

A right triangle has one angle that is exactly 90 degrees. This property is what makes it a 'right' triangle. The sides of a right triangle have special relationships with each other, especially the lengths of the sides.

Here's how these relationships work:

• The two sides forming the right angle are called 'legs'.
• The side opposite the right angle is the 'hypotenuse' and it is the longest side of the triangle.

The Pythagorean theorem plays a key role in figuring out side lengths. According to it, the square of the hypotenuse is equal to the sum of the squares of the two legs. Mathematically, this can be represented as:
\(a^2 + b^2 = c^2\)

Where:

• \(a\) is one leg.
• \(b\) is the other leg.
• \(c\) is the hypotenuse.

Understanding this setup is the starting point to solve problems involving a right triangle.

quadratic equation

A quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\), where $$a$$, $$b$$, and $$c$$ are constants (with $$a eq 0$$). Solving these equations often involves factoring, using the quadratic formula, or completing the square.

For our exercise, we needed to solve the equation:
\(x^2 - 200x - 150000 = 0\)
To solve it using the quadratic formula, we use:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, $$a$$ = 1, $$b$$ = -200, and $$c$$ = -150000. Plugging these values in, we get:
\[x = \frac{200 \pm \sqrt{200^2 + 600000}}{2}\]
\[x = \frac{200 \pm \sqrt{640000}}{2}\]
\[x = \frac{200 \pm 800}{2}\]

This results in two solutions, but only the positive one makes sense for this problem:\br>

• \(x = 500\) yards

Having a grasp on quadratic equations helps solve these kinds of geometric problems effectively.

walkway length calculation

To find the total length of the walkway, we must add up the lengths of all three sides of the right triangle.

We start by determining the lengths:

• Shorter leg = 500 yards
• Longer leg = 500 + 700 = 1200 yards
• Hypotenuse = 500 + 800 = 1300 yards

Adding these, the total length of the walkway is:
\[500 + 1200 + 1300 = 3000\] yards

This method leverages understanding of side relationships in right triangles and is the final step in solving the given problem. Combining these steps makes complex calculations simpler and ensures accuracy.