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Chapter 14: Problem 93

The power expended by a cheetah is \(160 \mathrm{kW}\) while running at $110 \mathrm{km} / \mathrm{h},$ but its body temperature cannot exceed \(41.0^{\circ} \mathrm{C} .\) If \(70.0 \%\) of the energy expended is dissipated within its body, how far can it run before it overheats? Assume that the initial temperature of the cheetah is \(38.0^{\circ} \mathrm{C},\) its specific heat is \(3.5 \mathrm{kJ} /\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right),\) and its mass is \(50.0 \mathrm{kg}\)

Short Answer

Expert verified
Answer: The cheetah can run approximately 0.65 km before it overheats.

Step by step solution

01

Calculate energy that can be absorbed by the cheetah

To find out how much energy the cheetah can absorb before overheating, we will use the specific heat formula, which is: \(Q = mc(T_f - T_i)\) Where: Q: energy that can be absorbed m: mass of the object c: specific heat capacity \(T_f\): final temperature \(T_i\): initial temperature Here, \(T_f = 41.0^{\circ} \mathrm{C},\) \(T_i = 38.0^{\circ} \mathrm{C},\) and \(c = 3.5 \mathrm{kJ/ (kg} \cdot^{\circ}\mathrm{C)},\) and \(m = 50.0 \mathrm{kg}\). Let's plug in these values and compute the absorbed energy (Q).
02

Calculate the energy released by the cheetah while running

We know that 70% of the energy is dissipated within the cheetah's body. Thus, to calculate the energy released while running, we multiply the power with the percentage of energy dissipated: \(E = P\cdot t \cdot 0.7\) Where: E: energy released by the cheetah P: power expended (= 160 kW) t: time for which the cheetah runs We need to find t and then the energy released will be equal to the absorbed energy (Q).
03

Find the time for which the cheetah runs

Using the absorbed energy (Q) and energy released (E), we can find the time (t) as: \(t = \frac{Q}{0.7\cdot P}\) Now let's substitute the values for Q and P to find t.
04

Find the distance covered by the cheetah while running

We know the speed of the cheetah is 110 km/h, and we've found the time (t) it takes before overheating. We can now find the distance covered using the formula: Distance = Speed \(\times\) Time Let's plug in the values for speed and time to find the distance covered before the cheetah overheats.

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Most popular questions from this chapter

A cylinder contains 250 L of hydrogen gas \(\left(\mathrm{H}_{2}\right)\) at \(0.0^{\circ} \mathrm{C}\) and a pressure of 10.0 atm. How much energy is required to raise the temperature of this gas to \(25.0^{\circ} \mathrm{C} ?\)
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