Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Answer:
a. 2in
b. [tex]l(t)=0.002t^{3}-0.06t^{2}+0.87t+2cos(\frac{\pi}{12}t)+38[/tex]
c. l'(3)=0.1938 in/hr
Step-by-step explanation:
a.
In part a, the problem is asking us to find how much snow will melt during the 6 hour period. In order to do so we need to take the M(t) function and integrate it in the given period of time, so we get:
[tex]\int\limits^6_0 {\frac{\pi}{6}sin(\frac{\pi}{12}t)} \, dt[/tex]
So now we solve:
[tex]\frac{\pi}{6}\int\limits^6_0 {sin(\frac{\pi}{12}t)} \, dt[/tex]
this is a known integral, so we get:
[tex]\frac{\pi}{6}[-\frac{12}{\pi}cos(\frac{\pi}{12}t)]^{6}_{0}[/tex]
we can simplify this so we get:
[tex]-2[cos(\frac{\pi}{12}t)]^{6}_{0}[\tex]
[tex]-2[cos(\frac{\pi}{12}(6))-cos(\frac{\pi}{12}(0))][/tex]
[tex]-2[cos(\frac{\pi}{2})-cos(0)][/tex]
[tex]-2[0-1][/tex]
2 in
b)
For part b they want us to write an expression for l(t), the total number of inches of snow at any time t.
in order to do so we need to find an expression for the rate of change of snow. A snowmaker is adding some snow while another amountn of snow is melting, so this rate of change is found by subtracting the two functions, so we get:
l'(t)=S(t)-M(t)
so:
[tex]l(t)=\int {(0.006t^{2}-0.12t+0.87-\frac{\pi}{6}sin(\frac{\pi}{12}t))} \, dt[/tex]
so we integrate this to get:
[tex]l(t)=\frac{0.006}{3}t^{3}-\frac{0.12}{2}t^{2}+0.87t+2cos(\frac{\pi}{12}t)+C[/tex]
and we can simplify this:
[tex]l(t)=0.002t^{3}-0.006t^{2}+0.87t+2cos(\frac{\pi}{12}t)+C[/tex]
so now we need to find what C is equal to, we can use the fact that l(0)=40in so we get:
[tex]40=0.002(0)^{3}-0.006(0)^{2}+0.87(0)+2cos(\frac{\pi}{12}(0))+C[/tex]
which yields:
40=2+C
so
C=40-2
C=38
So the final function is:
[tex]l(t)=0.002t^{3}-0.06t^{2}+0.87t+2cos(\frac{\pi}{12}t)+38[/tex]
c)
for part c we need to find the rate of change of the total amount of snow at time t=3, so in this case we can use the equation we found previously:
l'(t)=S(t)-M(t)
[tex]l'(t)=(0.006t^{2}-0.12t+0.87-\frac{\pi}{6}sin(\frac{\pi}{12}t))[/tex]
and substitute t=3
[tex]l'(3)=(0.006(3)^{2}-0.12(3)+0.87-\frac{\pi}{6}sin(\frac{\pi}{12}(3)))[/tex]
which yields:
l'(t)=0.1938 in/hr
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.