Problem 37 - Algebra and Trigonometry

Question

Mixture Problem The radiator in a car is filled with a solution of 60\(\%\) antifreeze and 40\(\%\) water. The manufacturer of the antifreeze suggests that, for summer driving, optimal cooling of the engine is obtained with only 50\(\%\) antifreeze. If the capacity of the radiator is 3.6 \(\mathrm{L}\) , how much coolant should be drained and replaced with water to reduce the antifreeze concentration to the recommended level?

Step-by-Step Solution

Answer

Drain 0.36 L of coolant and replace with water.

Step 1: Identify the Current Amount of Antifreeze

The radiator is filled with a 60% antifreeze solution and has a total capacity of 3.6 L. Therefore, the current amount of antifreeze in the radiator is \( 0.6 \times 3.6 = 2.16 \; \text{L} \) of antifreeze.

Step 2: Determine Desired Antifreeze Amount

To achieve a 50% antifreeze concentration in a 3.6 L radiator, the desired amount of antifreeze is \( 0.5 \times 3.6 = 1.8 \; \text{L} \).

Step 3: Calculate the Amount of Antifreeze to Remove

We currently have 2.16 L of antifreeze and desire only 1.8 L. The difference tells us how much antifreeze needs to be removed: \( 2.16 - 1.8 = 0.36 \; \text{L} \) of antifreeze needs to be drained.

Step 4: Replace Drained Antifreeze with Water

Since the volume needs to remain 3.6 L, the 0.36 L of antifreeze drained is replaced by 0.36 L of water. This keeps the volume constant while adjusting the antifreeze concentration to 50%.

Key Concepts

Antifreeze Concentration

Antifreeze concentration refers to the ratio of antifreeze present in the coolant mixture within a car’s radiator.
Antifreeze is crucial because it lowers the freezing point of the liquid in your radiator, preventing engine freezing in cold weather. It also raises the boiling point, enhancing heat absorption during warming. Typically, a mixture contains both water and antifreeze.
It's expressed as a percentage, indicating how much of the total fluid volume is made up of antifreeze. In this scenario, the initial concentration of antifreeze is 60%, while the desired concentration for optimized summer performance is 50%.
This adjustment may be necessary to ensure that the engine’s cooling system works effectively under different temperature conditions.

Radiator Capacity

Radiator capacity is the total volume of coolant that a car's radiator can hold. This capacity is key in ensuring the engine is consistently cooled and operates efficiently.
In this problem, the radiator has a fixed capacity of 3.6 liters, which means that the sum of water and antifreeze should not exceed this volume.
Understanding the radiator’s capacity helps in determining how much fluid can be replaced or needs to be added. By knowing the capacity, one can calculate precisely how much to drain and refill to adjust the concentration of fluids without overfilling or underfilling.

Coolant Replacement

Coolant replacement involves draining part of the radiator's existing fluid mixture and refilling it with a new composition to achieve a desired antifreeze-to-water ratio.
This is essential for maintaining engine performance and avoiding overheating or freezing of the engine components. To modify the antifreeze concentration, a precise amount of the current mixture must be drained and then replaced with water.
In this exercise, 0.36 L of the current coolant mixture is removed. This ensures that the radiator maintains its capacity while adjusting the antifreeze percentage, preventing damage due to excess or insufficient antifreeze.

Engine Cooling Optimization

Engine cooling optimization is all about ensuring the vehicle's cooling system is perfectly balanced to handle temperature variations. Proper cooling is critical for engine efficiency and longevity.
For optimal cooling, especially during summer, adjusting the antifreeze concentration as recommended by the manufacturer is vital. Too much antifreeze can be as problematic as too little.
This exercise shows how an appropriate mixture significantly influences the cooling capability, helping to prevent possible engine loss due to overheating or inefficiency.

• 50% antifreeze is suggested for effective heat transfer.
• Keeping the cooling system in tip-top shape prevents repairs.

Making such adjustments ensures that your engine stays cool under the pressure of high temperatures or long drives.