Problem 15 - Intermediate Algebra

Question

Farming. In 10 minutes, a conveyor belt can move \(1,000\) bushels of corn into the storage bin shown. A smaller belt can move \(1,000\) bushels to the storage bin in 14 minutes. If both belts are used, how long will it take to move \(1,000\) bushels to the storage bin?

Step-by-Step Solution

Answer

It will take approximately 5.83 minutes to move 1,000 bushels with both belts.

Step 1: Calculate the Rate of the First Belt

The rate of the first conveyor belt can be calculated by determining how many bushels it moves per minute. Since it moves 1,000 bushels in 10 minutes, its rate is \( \frac{1000}{10} = 100 \) bushels per minute.

Step 2: Calculate the Rate of the Second Belt

Similarly, calculate the rate of the smaller conveyor belt. It moves 1,000 bushels in 14 minutes, so the rate is \( \frac{1000}{14} \approx 71.43 \) bushels per minute.

Step 3: Determine the Combined Rate

Now, find the total rate when both belts are operating together. This is done by adding the individual rates: \(100 + 71.43 = 171.43\) bushels per minute.

Step 4: Calculate the Time Required Together

To find the time required to move 1,000 bushels with this combined rate, use the formula \( \text{Time} = \frac{\text{Total Work}}{\text{Rate}} \). Therefore, \( \text{Time} = \frac{1000}{171.43} \approx 5.83 \) minutes.

Key Concepts

Conveyor Belt

A conveyor belt in mechanical systems can be visualized as a moving walkway for products. It’s a simple and efficient technology used in numerous industries, from farming to manufacturing. In the provided exercise, we focus on a conveyor belt that transports bushels of corn.
The primary function here is to automate the movement of corn from the field into storage. Imagine standing at a grocery store checkout line, where the conveyor belt moves your groceries toward the cashier.
Similarly, these belts move bushels efficiently to a storage bin. The crux of understanding how a conveyor belt contributes to work rate problems lies in knowing its capacity and speed. Just like any transportation system, the efficiency of a conveyor belt is measured by how quickly it can complete its job—here, in terms of bushels moved per minute. Thus, knowing each belt's rate directly helps determine how efficiently they can work alone or in tandem.

Combined Rate

Work rate problems often involve combining rates from multiple sources to achieve a task faster. In this case, we combine the rates of both conveyor belts to move a set amount of bushels. **Understanding Combined Rates:**

• Each conveyor belt performs an independent rate of work, which we calculate as bushels per minute.
• To find the combined rate, simply add the rates of the individual belts together.

By adding the rates of the two conveyor belts, we determine how quickly they can complete the work when used together. This problem illustrates how using two systems simultaneously can significantly reduce the time required to accomplish a task. For instance, in our exercise, a combined rate of roughly 171.43 bushels per minute means that together, the belts move corn to storage more than just the sum of their individual efforts.
It becomes a powerful shortcut to solving problems quickly when time matters, as it often does in real-world operations.

Bushels per Minute

The unit of measure in this exercise is bushels per minute, which illustrates the efficiency of the conveyor belts. This measurement tells us how many bushels are moved within a specific time frame. **Why Bushels per Minute Matter:**

• This unit allows for a direct comparison between different conveyor belts.
• It's a standard rate that shows productivity.
• Helps in calculating the total time to complete a given task when multiple systems are working.

Calculating bushels per minute involves dividing the total bushels moved by the time it takes to move them. For example, for the first belt in our problem, we see it moves 100 bushels per minute. This not only indicates speed but also helps project how long a task will take if this rate remains constant. Understanding this concept is essential in any scenario where different systems with different efficiencies must be compared or combined to optimize workloads.