The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function
[tex]$D(t)=4 \cos \left(\frac{\pi}{4} t+\frac{9 \pi}{4}\right)+2$[/tex]
where [tex]$t$[/tex] is the number of hours after midnight. Find the rate at which the depth is changing at 5 a.m. Round your answer to 4 decimal places and make sure your calculator is in radian mode.
[tex]$\square$[/tex] ft/hour
Answer (1)
Find the derivative of the depth function D ( t ) with respect to t using the chain rule: D ′ ( t ) = − " , " p i " , " s in " , " l e f t ( " , " f r a c " , " p i 4 t + " , " f r a c 9" , " p i 4 " , " r i g h t ) .
Evaluate the derivative at t = 5 : D ′ ( 5 ) = − " , " p i " , " s in " , " l e f t ( " , " f r a c 14" , " p i 4 " , " r i g h t ) = " , " p i .
Round the result to 4 decimal places: " , " p i ≈ 3.1416 .
The rate at which the depth is changing at 5 a.m. is 3.1416 ft/hour.
Explanation
Problem Analysis We are given the depth function D ( t ) = 4" , " cos " , " l e f t ( " , " f r a c " , " p i 4 t + " , " f r a c 9" , " p i 4 " , " r i g h t ) + 2 and asked to find the rate at which the depth is changing at 5 a.m. This means we need to find the derivative of D ( t ) and evaluate it at t = 5 .
Finding the Derivative First, we find the derivative of D ( t ) with respect to t . Using the chain rule, we have
D ′ ( t ) = − 4" , " s in " , " l e f t ( " , " f r a c " , " p i 4 t + " , " f r a c 9" , " p i 4 " , " r i g h t ) " , " c d o t " , " f r a c " , " p i 4 = − " , " p i " , " s in " , " l e f t ( " , " f r a c " , " p i 4 t + " , " f r a c 9" , " p i 4 " , " r i g h t ) .
Evaluating the Derivative Next, we evaluate the derivative at t = 5 :
D ′ ( 5 ) = − " , " p i " , " s in " , " l e f t ( " , " f r a c " , " p i 4 ( 5 ) + " , " f r a c 9" , " p i 4 " , " r i g h t ) = − " , " p i " , " s in " , " l e f t ( " , " f r a c 14" , " p i 4 " , " r i g h t ) = − " , " p i " , " s in " , " l e f t ( " , " f r a c 7" , " p i 2 " , " r i g h t ) .
Since " , " s in " , " l e f t ( " , " f r a c 7" , " p i 2 " , " r i g h t ) = − 1 , we have
D ′ ( 5 ) = − " , " p i ( − 1 ) = " , " p i .
Rounding the Result Finally, we round the result to 4 decimal places. Since " , " p i ≈ 3.14159265 , rounding to 4 decimal places gives us 3.1416.
Final Answer The rate at which the depth is changing at 5 a.m. is approximately 3.1416 ft/hour.
Examples
Understanding rates of change is crucial in many real-world applications. For example, consider a marine biologist studying the population growth of a certain species of fish in a controlled environment. By modeling the population size as a function of time, the biologist can use calculus to determine the rate at which the population is growing or shrinking at any given moment. This information is vital for making informed decisions about resource management and conservation efforts, ensuring the long-term health and sustainability of the ecosystem.
