Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

As shown on the following page, a wheelbarrow is being wheeled down a perfectly straight 13 -foot ramp. The top of the ramp is 5 feet high and the ramp covers a horizontal distance of 12 feet. When the wheelbarrow has moved a horizontal distance of 6 feet it is moving horizontally at a rate of \(0.5\) feet per second. At what rate is it moving vertically downward?

Step-by-Step Solution

Answer

The wheelbarrow is moving vertically downward at a rate of approximately \(0.2083\) feet per second.

Step 1: Calculate the Slant Distance of the Ramp

Use the Pythagorean theorem to calculate the slant distance of the ramp. The theorem says the hypotenuse (slant distance in this case) equals the square root of the sum of the squares of the other two sides. Hence, \(\sqrt{5^2 + 12^2} = 13\) feet is the slant distance of the ramp.

Step 2: Calculate the Vertical to Horizontal Ratio

Use the given vertical distance and the calculated slant distance to find the ratio of vertical distance to slant distance. This comes out as \(\frac{5}{13}\). Similarly, the horizontal to slant distance ratio is \(\frac{12}{13}\). The vertical to horizontal ratio, which will be required in the next step, is the division of these two, equal to \(\frac{5}{12}\).

Step 3: Calculate Vertical Speed

Multiply the horizontal speed (0.5 feet per second) by the vertical to horizontal ratio calculated in the previous step to find the vertical speed using the given similarity in the right triangle, \(\frac{5}{12} * 0.5 = 0.2083\) feet per second. The wheelbarrow is moving vertically downward at this rate.

Key Concepts

Understanding the Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry, crucial for solving problems involving right triangles. In essence, it explains how the three sides of a right triangle are related. According to this theorem, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides, which are the adjacent and opposite sides. In mathematical terms, if we denote the hypotenuse by 'c' and the other two sides by 'a' and 'b', the theorem is expressed as:
\[c^2 = a^2 + b^2\].
In the context of related rates problems, especially those involving geometric motion, the Pythagorean theorem helps us find missing distances or establish relationships between varying quantities. For example, in our wheelbarrow problem, by knowing the horizontal and vertical sides of the ramp, we can use the Pythagorean theorem to establish the length of the ramp, which directly impacts the rate at which the wheelbarrow descends vertically.

Velocity Ratio in Related Rates

The concept of velocity ratios becomes invaluable when dealing with related rates problems in calculus. When two or more quantities are changing over time and are linked geometrically, their rates of change can be related through a velocity ratio. In our wheelbarrow scenario, we are given the horizontal speed and asked to find the vertical speed. To establish this relationship, we observe that the wheelbarrow's movement along the ramp creates a right triangle, with the hypotenuse being the ramp itself and the other two sides representing the horizontal and vertical distances covered.
Calculating the ratio of these distances helps us form a proportion that relates the horizontal and vertical speeds. By dividing the vertical distance by the horizontal distance (or vice versa), we obtain a ratio that, when multiplied by the known horizontal speed, yields the rate at which the wheelbarrow descends vertically. This method leverages the geometric relationship between the sides of a triangle and the respective velocities, allowing us to solve complex dynamical systems with relative ease.

Calculating Vertical Speed

Calculating vertical speed in related rates problems is a practical application of calculus that combines geometric understanding with the principles of differentiation. When an object moves along an inclined path, as our wheelbarrow does along the ramp, its horizontal and vertical motions are interconnected. The vertical speed can be determined by multiplying the horizontal speed by the aforementioned vertical to horizontal ratio obtained from geometric relations—in this case, a right triangle formed by the ramp.
In simpler terms, if we know the speed at which the wheelbarrow is moving horizontally (\(0.5\) feet per second), we can find out how fast it is descending vertically by relating the two speeds. This is done through the velocity ratio we derived earlier (\(\frac{5}{12}\)). By multiplying this ratio by the horizontal speed, we find the vertical speed to be \(0.5 \times \frac{5}{12} = 0.2083\) feet per second. This step-by-step process reveals how the rates of change in different directions can inform each other when objects are in motion.