Math Problem Statement

As a block of ice melts, its weight changes from 13/4 lb to 4 lb. This change takes 1/4 h. The ice melts at a constant rate. At what rate does the weight of the ice change? Show your work.

Solution

The problem is asking us to determine the rate of change in the weight of a block of ice as it melts, given the initial and final weights and the time taken for the change.

Information provided:

  • Initial weight: 134lb\frac{13}{4} \, \text{lb}
  • Final weight: 4lb4 \, \text{lb}
  • Time taken for the change: 14h\frac{1}{4} \, \text{h}

Step 1: Calculate the weight change.

Weight change=Final weightInitial weight\text{Weight change} = \text{Final weight} - \text{Initial weight} Weight change=4134\text{Weight change} = 4 - \frac{13}{4}

Convert 44 into a fraction with a denominator of 4: 4=1644 = \frac{16}{4}

Now subtract: Weight change=164134=34lb\text{Weight change} = \frac{16}{4} - \frac{13}{4} = \frac{3}{4} \, \text{lb}

Step 2: Calculate the rate of change.

The rate of change is the weight change divided by the time: Rate of change=Weight changeTime\text{Rate of change} = \frac{\text{Weight change}}{\text{Time}}

Substitute the values: Rate of change=3414\text{Rate of change} = \frac{\frac{3}{4}}{\frac{1}{4}}

Dividing fractions: Rate of change=34×41=3lb/h\text{Rate of change} = \frac{3}{4} \times \frac{4}{1} = 3 \, \text{lb/h}

Final Answer:

The rate of change in the weight of the ice is: 3lb/h\boxed{3 \, \text{lb/h}}

Let me know if you want further details or clarification!


Here are 5 related questions to explore:

  1. What would the rate of change be if the ice took 12h\frac{1}{2} \, \text{h} to melt instead?
  2. How would you calculate the weight of the ice after a certain amount of time, assuming it melts at this constant rate?
  3. What is the total time required for the block of ice to completely melt if it starts at 134lb\frac{13}{4} \, \text{lb}?
  4. How does the rate of change differ if the melting process happens in a shorter or longer time?
  5. Can we use this rate to determine the remaining weight after any given time? How?

Tip:

When working with fractions, always simplify before performing operations to avoid unnecessary complications.

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Math Problem Analysis

Mathematical Concepts

Rate of Change
Fraction Operations
Basic Algebra

Formulas

Rate of Change = Change in Weight / Change in Time

Theorems

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Suitable Grade Level

Grades 6-8