A washing machine's "warm" setting calls for water at 34.0^∘ C. If the cold-water supply is at 1

step 1 In this problem we are asked to calculate the ratio of hot to cold water in the washing machine.

A washing machine's "warm" setting calls for water at 34.0^∘...

A washing machine's "warm" setting calls for water at $34.0^{\circ} \mathrm{C}$. If the cold-water supply is at $12.4^{\circ} \mathrm{C}$ and the hot-water supply is at $51.7^{\circ} \mathrm{C},$ what ratio of hot to cold water should the washing machine's fill valves admit to the machine?

Key Concepts

Ratio Calculation

Calculating the ratio of inputs in a mixture problem involves setting up an equation that reflects the balance between the contributions of each component. In this context, the ratio of hot water to cold water is determined by how much each water temperature differs from the final desired temperature, ensuring that the mixture achieves the target temperature.

Conservation of Energy

This principle states that in the absence of heat loss, the total thermal energy of a closed system remains constant during mixing. When two streams of water with different temperatures are combined, the heat lost by the hotter water is equal to the heat gained by the cooler water, leading to a final equilibrium temperature that is a weighted average of the two.

Weighted Average

This concept refers to the idea that the final temperature of a mixture is determined by the proportionate contributions of the individual components, each weighted by its mass or volume. In mixing problems, the overall temperature is calculated by adding the products of the temperature and the mass or volume of each component, and then dividing by the total mass or volume.

Linear Interpolation

Linear interpolation is used to estimate an unknown value within two known values. In the context of mixing water of different temperatures, it provides a method to determine the necessary fraction of hot or cold water needed to achieve a specific target temperature by assuming that the temperature response is linear between the two supplies.

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