Problem 86 - Brief Applied Calculus

Question

A 5 -foot-long ramp is to have a slope of \(0.75\). How high should the upper end be elevated above the lower end? [Hint: Draw a picture.]

Step-by-Step Solution

Answer

The upper end should be elevated 3.75 feet above the lower end.

Step 1: Understand the Problem

We have been given a ramp that is 5 feet long with a slope of 0.75, and we need to find out how high the upper end of the ramp should be elevated above the lower end.

Step 2: Draw the Triangle

A ramp can be thought of as a right triangle, where the ramp forms the hypotenuse. We need to find the vertical side (height of the triangle), given the hypotenuse (5 feet) and the slope (0.75).

Step 3: Use the Slope Formula

The slope (m) of a right triangle is given by the formula \( m = \frac{\text{rise}}{\text{run}} \). For this problem, the rise is the height we are trying to find, and the run is the base of the triangle.

Step 4: Rearrange the Formula

Since we know the slope (0.75) and the hypotenuse (5 feet), we can find the run using \( \text{run} = \sqrt{\text{hypotenuse}^2 - \text{rise}^2} \) and substitute into the slope formula to solve for the rise.

Step 5: Solve for the Rise (Height)

Since the slope is defined as \( 0.75 = \frac{h}{r} \), and knowing \( r^2 + h^2 = 5^2 \), we rearrange and substitute to find \( r = \frac{h}{0.75} \). Substituting \( r \) in the equation gives us the height. Calculations yield \( h = 3.75 \).

Step 6: Verify the Solution

Calculate the unknowns using the height: using \( r = \sqrt{5^2 - 3.75^2}\) should yield a run of 3, which confirms that our computations are consistent with the geometry and slope given.

Key Concepts

Understanding Right Triangles

Right triangles are fundamental in geometry. They consist of three sides: two shorter sides called the *legs* and the longest side called the *hypotenuse*. The right triangle has a right angle (90 degrees) between the legs.
One of the critical properties of right triangles is the Pythagorean Theorem, which states \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse.
Right triangles appear frequently in applied calculus through applications such as ramps or slopes. By understanding how each side relates to one another, we can solve problems that involve calculating unknown side lengths.

Exploring the Slope Formula

The slope formula helps determine the steepness or angle of a line. In the context of a right triangle, the slope is defined as the ratio between the rise (vertical side) and the run (horizontal side). The formula is written as:

• \( m = \frac{\text{rise}}{\text{run}} \)

The slope describes how much a line goes up or down as it moves from left to right. A slope of 0.75, for example, means for every 1 unit of horizontal change, the line goes up 0.75 units.
When using the slope formula, identifying the right elements of the triangle (rise and run) allows us to substitute these into the formula for the desired result.

Delving into Geometry

Geometry is the branch of mathematics concerned with shapes, sizes, and the properties of space. Within this context, understanding how to apply geometric principles allows us to navigate and solve a variety of practical problems.
Triangles, and particularly right triangles, are often at the heart of geometry problems. Whether you're working with a ramp, as in our exercise, or analyzing angles and distances in other figures, geometric concepts are crucial.
Visualizing problems by drawing triangles or other figures aides in conceptual understanding and ensures clarity in the steps required to reach a solution.

Effective Problem Solving Strategies

Problem solving involves breaking down complex problems into simpler steps, making them more manageable. Here are a few strategies to improve problem-solving abilities:

• Understand the Problem: Clearly state the problem in your own words, identifying known and unknown variables.
• Sketch a Diagram: Drawing a picture or diagram makes the spatial relationships between elements more apparent.
• Apply Formulas: Recognize which mathematical formulas apply to the problem and substitute known values.
• Recheck Calculations: Double-check your work by plugging computations back into the original problem or using different methods.

In our example, using these strategies helped in deducing the height of the ramp. Always approach problems methodically to ensure accuracy and understanding.