Problem 19 - Contemporary Precalculus

Question

In the remaining exercises, solve the applied problems. A 13 -foot-long ladder leans on a wall, as shown in the figure. The bottom of the ladder is 5 feet from the wall. If the bottom is pulled out 3 feet farther from the wall, how far does the top of the ladder move down the wall? [Hint: Draw pictures of the right triangle formed by the ladder, the ground, and the wall before and after the ladder is moved. In each case, use the Pythagorean Theorem to find the distance from the top of the ladder to the ground.]

Step-by-Step Solution

Answer

Answer: The top of the ladder moves approximately 1.75 feet down the wall.

Step 1: Draw the initial right triangle

Draw a right triangle with the ladder as the hypotenuse, the ground as the base, and the wall as the height. The ladder is initially 13 feet long, and the distance between the wall and the bottom of the ladder is 5 feet. Label these distances and recognize that the height we are looking for is the distance from the top of the ladder to the ground.

Step 2: Use the Pythagorean Theorem to find the initial height

We know that the ladder is 13 feet long, and the base is 5 feet. Use the Pythagorean Theorem: a^2 + b^2 = c^2, where 'a' and 'b' are the legs of the right triangle (base and height), and 'c' is the hypotenuse (the ladder). In this case: (5)^2 + b^2 = (13)^2 25 + b^2 = 169 b^2 = 144 b = 12 Therefore, the initial height of the ladder on the wall is 12 feet.

Step 3: Draw the new right triangle and find the new height

Now, pull out the bottom of the ladder 3 feet farther from the wall, making the new distance 8 feet from the wall. Draw and label the new right triangle with the same ladder length of 13 feet, the new base of 8 feet, and the new unknown height.

Step 4: Use the Pythagorean Theorem to find the new height

Again, use the Pythagorean Theorem to find the new height: (8)^2 + b^2 = (13)^2 64 + b^2 = 169 b^2 = 105 b = sqrt(105) ≈ 10.25 The new height of the ladder on the wall is approximately 10.25 feet.

Step 5: Calculate the distance the top of the ladder moves down the wall

Now that we have both the initial height and the new height, we can find the difference between them to determine the distance the top of the ladder moves down the wall: 12 - 10.25 ≈ 1.75 The top of the ladder moves approximately 1.75 feet down the wall.

Key Concepts

Right Triangle

Understanding a right triangle is essential for solving problems using the Pythagorean Theorem. A right triangle is a type of triangle that has one angle exactly equal to 90 degrees. This types of triangle is composed of three sides:

• The hypotenuse: the longest side, always opposite the right angle.
• The base and the height: the two sides that form the right angle, also referred to as the "legs" of the triangle.

In our ladder problem, the ladder serves as the hypotenuse of the right triangle, the distance from the wall is the base, and the height is how far up the wall the ladder reaches. By sketching an image of these triangles, it becomes easier to visualize how changes in one dimension affect another. This visualization makes calculations more intuitive and straightforward.

Distance Calculation

Calculating distances in a right triangle often involves the Pythagorean Theorem, which is fundamental in geometry. This theorem applies specifically to right triangles and is expressed as: \[ a^2 + b^2 = c^2 \] where '\[a\]' and '\[b\]' are the legs of the triangle, and '\[c\]' is the hypotenuse. In the ladder scenario, we initially have:

• Base (\(a\)) = 5 feet
• Ladder (hypotenuse, \(c\)) = 13 feet

To find the initial height, we solve the equation: \[ 5^2 + b^2 = 13^2 \]resulting in \[ b = 12 \] feet, so the ladder reaches 12 feet up the wall. After moving the ladder and increasing the base to 8 feet, the new height is found similarly: solving\[ 8^2 + b^2 = 13^2 \]provides\[ b \approx 10.25 \]feet. The change in height is then the key to understanding how far the top of the ladder moves.

Geometry Application

The application of geometry in real-world contexts, like a ladder leaning against a wall, demonstrates how abstract mathematical principles can solve practical problems. In this exercise, visualizing the ladder's movement as triangles allows us to assess alterations in dimensions and calculate new parameters comfortably. By using the Pythagorean Theorem, we can predict the move of the ladder's top end with simple algebra:

• Before the movement: Height = 12 feet
• After the movement: Height = 10.25 feet

Hence, the ladder's apex descends by approximately 1.75 feet. This exercise not only highlights lateral and vertical distances but enriches our understanding of how geometry aids our comprehension of spaces, distances, and even the behavior of objects. Knowing how to judiciously apply such concepts is instrumental in a broad array of scientific and engineering fields.