For a cheetah, \(70.0 \%\) of the energy expended during exertion is internal work done on the cheetah's system and is dissipated within his body; for a dog only \(5.00 \%\) of the energy expended is dissipated within the dog's body. Assume that both animals expend the same total amount of energy during exertion, both have the same heat capacity, and the cheetah is 2.00 times as heavy as the dog. (a) How much higher is the temperature change of the cheetah compared to the temperature change of the dog? (b) If they both start out at an initial temperature of \(35.0^{\circ} \mathrm{C},\) and the cheetah has a temperature of \(40.0^{\circ} \mathrm{C}\) after the exertion, what is the final temperature of the dog? Which animal probably has more endurance? Explain.

Step by step solution

01

Calculate the percentage of energy that isn't dissipated within the animals' bodies

We are given that 70% of the energy is dissipated within the cheetah's body, so the percentage of energy that didn't dissipate internally is \((100 - 70)\% = 30\%\). Similarly, for the dog, only 5% of the energy is dissipated within his body, so the percentage of energy that didn't dissipate internally is \((100 - 5)\% = 95\%\).

02

Determine the ratio of energy dissipated within the animals

We can now determine the ratio of energy that's getting dissipated within the cheetah and the dog by dividing their energy dissipation percentages, which yields the following: \(\frac{70 \%}{5 \%} = 14\). This means that the cheetah's internal energy is 14 times higher than the dog's internal energy.

03

Calculate the ratio of temperature change in the cheetah and the dog

Since the cheetah is 2 times heavier than the dog and the heat capacities are the same for both, the ratio of the temperature change is equal to the ratio of energy dissipated. Therefore, the ratio of temperature change (ΔT) for the cheetah and the dog is: \(\frac{\Delta T_{cheetah}}{\Delta T_{dog}} = 14\)

04

Find the temperature change of the dog

We are given the initial temperature (\(T_{initial}=35^{\circ} \text{C}\)) and final temperature (\(T_{final, cheetah}=40^{\circ} \text{C}\)) of the cheetah. So, the temperature change of the cheetah is \(\Delta T_{cheetah} = T_{final, cheetah} - T_{initial} = 40^{\circ} \text{C} - 35^{\circ} \text{C} = 5^{\circ} \text{C}\). Now, using the ratio of temperature change found in step 3, we can find the temperature change of the dog as follows: \(\Delta T_{dog} = \frac{\Delta T_{cheetah}}{14} = \frac{5^{\circ} \text{C}}{14} \approx 0.357^{\circ} \text{C}\).

05

Calculate the final temperature of the dog

To find the final temperature of the dog, add the temperature change found in step 4 to the initial temperature: \(T_{final, dog} = T_{initial} + \Delta T_{dog} = 35^{\circ} \text{C} + 0.357^{\circ} \text{C} \approx 35.357^{\circ} \text{C}\).

06

Determine which animal has more endurance

Since the dog has a smaller increase in temperature, it dissipates a smaller percentage of energy internally and can more efficiently use its energy during exertion. Therefore, the dog probably has more endurance. The final temperature of the dog is approximately \(35.357^{\circ} \text{C}\), and the dog most likely has more endurance due to its lower internal energy dissipation.

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