Heat Efficiency of a Carnot Engine

[castrodisastro]

Homework Statement

A Carnot engine operates between a warmer reservoir at a temperature T₁ and a cooler reservoir at a temperature T₂. It is found that increasing the temperature of the warmer reservoir by a factor of 2.00 while keeping the same temperature for the cooler reservoir increases the efficiency of the Carnot engine by a factor of 5.00. Find the efficiency of the engine and the ratio of the temperatures of the two reservoirs in their original form.

Homework Equations

ε=1-(T₂/T₁)

The Attempt at a Solution

I think we are supposed to get a numerical value for an answer but I don't think I have enough information.

So I start with

ε=1-(T₂/T₁)

The question states, "increasing the temperature of the warmer reservoir by a factor of 2.00 while keeping the same temperature for the cooler reservoir increases the efficiency of the Carnot engine by a factor of 5.00"

That means T₁ will be doubled and once that happens, ε is quintupled, resulting in...

5ε=1-(T₂/2T₁)

The question asks for the efficiency, which I am supposed to answer with a numerical value. So I tried solving for ε.

ε=(1/5)-(T₂/10T₁))

I can't think of a way to solve this with 2 independent variables. Am I ignoring something from the question that would point me towards a value? Maybe I'm forgetting some mathematical property?

Once I have the efficiency then I can solve for the ratio of the two temperatures. Since it asks me for the ratio, I assume that knowing T₁ and T₂ are not required, so how could we solve for ε and get a numerical value?

Thanks in advance

 

[BvU]

Use R for $T_1/T_2$. Now you have 2 equations in 2 unknowns!

 

[castrodisastro]

Use R for $T_1/T_2$. Now you have 2 equations in 2 unknowns!

Thanks! I knew it was staring at me in the face.