Problem 129 - College Algebra Essentials

Question

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Step-by-Step Solution

Answer

The maximum number of bags of cement that can be safely lifted on the elevator in one trip is 28 bags.

Step 1: Identify The Problem

Firstly, the verbal conditions of the problem should be analyzed. It's a word problem that involves weights and a maximum capacity. To find the solution for this problem it is best to outline the known quantities first: The maximum capacity of the elevator is 3000 pounds. The elevator operator and each bag of cement weigh 245 pounds and 95 pounds respectively.

Step 2: Define The Variables

Let's define the variable:Let \( x \) be the number of bags of cement that can be lifted on the elevator in one trip.

Step 3: Formulate The Inequality

According to the problem, the total weight of the bags of cement plus the weight of the elevator operator must not exceed the maximum capacity of the elevator:Thus, the inequality is defined as: \( 245 + 95x \leq 3000 \). This is because the weight of each bag multiplied by the number of bags, added to the weight of the operator must be less than or equal to the maximum elevator weight.

Step 4: Solve The Inequality

Now, for solving the inequality: \( 95x \leq 3000 - 245 \), or: \( 95x \leq 2755\), which on further simplification gives approximately: \( x \leq 28.95 \). Since the number of bags of cement must be a whole number, we round down to get the maximum number of bags as \(x \leq 28\).

Step 5: Check the Solution

A final step would be to confirm our answer. Substituting \(x = 28\) into our inequality: \( 245 + 95*28 = 2895 \), which is less than or equal to 3000. So, this confirms that the solution is correct.