Problem 48 - Algebra 1

Question

In baseball the pitcher's mound is 60.5 feet from home plate. The strike zone, or distance across the plate, is 17 inches. The time it takes for a baseball to reach home plate can be determined by dividing the distance the ball travels by the speed at which the pitcher throws the baseball. If a pitcher throws a baseball at 90 miles per hour, how many seconds does it take for the baseball to reach home plate?

Step-by-Step Solution

Answer

The baseball takes approximately 0.45 seconds to reach home plate.

Step 1: Convert Units

Firstly, to ensure all units are consistent, convert the speed of the baseball from miles per hour to feet per second. We know that 1 mile equals 5280 feet and 1 hour equals 3600 seconds, so the conversion would be \(90 \times \dfrac{5280}{3600}\) feet per second.

Step 2: Use the Formula

Next, we use the formula to find the time. The formula is \(time = \dfrac{distance}{speed}\). Substitute in the distance of 60.5 feet and the converted speed.

Step 3: Compute the Answer

Finally, do the division to find the time it takes the baseball to reach home plate. This will be our answer in seconds.

Key Concepts

Unit Conversion

Grasping the concept of unit conversion is crucial for solving real-world problems in science, engineering, and everyday life. Unit conversion is the process of converting a measure from one unit to another without changing the quantity. For example, in the exercise, we need to convert the pitcher's throw speed from miles per hour to feet per second to make it compatible with the distance measured in feet.

Performing this conversion requires knowing the conversion factors: 1 mile is equivalent to 5280 feet, and 1 hour comprises 3600 seconds. The arithmetic involves multiplying the speed (90 mph) by the ratio of these conversion factors, resulting in speed in feet per second. Always ensure that you cancel out the original units correctly to end up with the desired units.

Rate Calculations

In many scenarios, including sports and physics, understanding rate calculations is fundamental. A rate is simply how fast something happens or changes over time. In this case, the rate is the speed of a baseball pitch, expressed as a distance covered over a specific time period.

In our exercise, after converting the units to make them consistent, we calculate the rate at which the baseball travels to home plate. Once we have that consistent rate in feet per second, we can use it to find out how long it takes for the ball to travel the distance of 60.5 feet from the pitcher's mound to the plate. It's essential to grasp that when units are consistent, rate calculations are straightforward: distance divided by speed gives you time.

Distance Time Speed Relationship

The relationship between distance, time, and speed is a fundamental concept in various fields such as physics and transportation. It's encapsulated in the formula: \[ time = \dfrac{distance}{speed} \].

This equation tells us that the time taken to travel a certain distance is determined by dividing the distance by the speed at which one travels. If any two of the variables are known, the third can be calculated. In baseball, knowing how long it takes for a pitch to reach home plate is useful for batters and coaches alike. By substituting the distance from the pitcher's mound to home plate, which is 60.5 feet, and the converted speed of the ball in feet per second into this formula, we can calculate the exact time in seconds that the ball takes to reach the batter's box. Understanding this relationship is crucial for predicting outcomes in dynamic and fast-paced environments.