Question

The time \(t\) (in seconds) it takes for sound to travel 1 kilometer can be modeled by $$ t=\frac{1000}{0.6 T+331} $$ where \(T\) is the air temperature (in degrees Celsius). a. You are 1 kilometer from a lightning strike. You hear the thunder \(2.9\) seconds later. Use a graph to find the approximate air temperature. b. Find the average rate of change in the time it takes sound to travel 1 kilometer as the air temperature increases from \(0^{\circ} \mathrm{C}\) to \(10^{\circ} \mathrm{C}\).

Short answer

The air temperature at the time of the lightning strike was approximately 24 degrees Celsius. The average rate of change in the time it takes for the sound to travel 1km as the air temperature increases from \(0^{\circ}C\) to \(10^{\circ}C\) is -0.0232 seconds per degree Celsius.

Step-by-step solution

Solve for air temperature

To find the air temperature for a travel time of 2.9 seconds, set \(t=2.9\) and solve for \(T\): \[2.9 = \frac{1000}{0.6 T + 331}\]To solve for \(T\), first cross-multiply\[2.9(0.6T + 331) = 1000\]Then, simplify the equation and solve for \(T\)\[T = \frac{1000}{2.9} - \frac{331}{0.6}\]

Calculate the approximate air temperature

By substituting the values into the equation, we get\[T \approx 23.96 ^{\circ}C\]This means the air was approximately 24 degrees Celsius when the lightning struck.

Find the average rate of change

The average rate of change is found by using the formula \((f(b)-f(a)) / (b-a)\). Here, we are to find the average rate of change in the time it takes for sound to travel 1km as the air temperature increases from \(0^{\circ}C\) to \(10^{\circ}C\). Thus, evaluate the function \(t = \frac{1000}{0.6 T + 331}\) at \(T = 0\) and \(T = 10\). Find \(f(b) = \frac{1000}{0.6 * 10 + 331}\) and \(f(a) = \frac{1000}{0.6 * 0 + 331}\). Use these values in the average rate of change formula.

Calculate the average rate of change

After evaluating the above, obtain \(f(b) \approx 2.789\) and \(f(a) \approx 3.021\). Now, substitute these in the average rate of change formula:\[\frac{f(b)-f(a)}{b-a} = \frac{2.789 - 3.021}{10 - 0} = -0.0232\]This means the average rate of change in time as temperature increases from \(0^{\circ}C\) to \(10^{\circ}C\) is around \(-0.0232\) seconds per degree Celsius.

Key concepts

Rate of Change in Acoustics

In acoustics, the rate of change refers to how quickly sound properties, such as its speed, are affected by variables like temperature. To understand this, imagine you're outside on a volcanic day. The rate at which sound travels to you from a distant drumbeat can actually vary with the air's warmth. When we calculate the average rate of change in acoustics, we're measuring the effect temperature has on sound travel time over a specific temperature interval.

The concept of rate of change is critical in understanding how environmental factors impact sound. For our exercise, we calculated it for a 1 kilometer sound travel as air temperature rose from 0°C to 10°C. To find it, we used a function of time against temperature and found that the time it takes for sound to travel decreases by approximately 0.0232 seconds for each degree Celsius increase. This is a subtle but noticeable phenomenon in real-world scenarios, such as estimating distances during a thunderstorm based on the delay between lightning and thunder.

Solving Algebraic Equations

Algebraic equations are mathematic bread and butter – we use them to describe relationships and solve for unknowns. In our example, we had an equation that modeled sound's travel time based on air temperature. To determine the air temperature after hearing thunder 2.9 seconds later, we cleverly rearranged the equation and solved for the temperature variable, T.

This process highlights a fundamental algebra skill: manipulating equations to isolate and solve for one variable. It's a dance of adding, subtracting, multiplying, and dividing both sides until the variable stands alone. Here, we moved things around and used cross-multiplication that allowed us to find that the air temperature at the time of the lightning strike was approximately 24°C. Mastering these steps is key to unlocking a wide range of problems, in math and beyond!

Temperature Effects on Sound Speed

Air temperature is more than just a number on a thermometer; it's a key player in the speed at which sound waves travel. The warmer the air, the faster sound waves jiggle through it. This is because heat energy causes air molecules to vibrate more briskly, paving the way for sound to zip through quicker.

In our exercise, by observing the travel time of sound over a kilometer, we uncovered the clear link between temperature and sound speed. With the equation given, we could debunk the puzzle of temperature's impact on sound, finding it speeds up slightly as the temperature climbs from icier to toasty levels. As we wrapped our brains around it and toyed with the equation, we saw firsthand the interplay between temperature and the haste of sound's journey.