Let P(t)=100+20 cos⁡ 6t,0leq tleq frac{pi}{2}. Find the maximum and minimum values for P, if any.

Question
Functions
asked 2020-10-26
Let \(P(t)=100+20 \cos⁡ 6t,0\leq t\leq \frac{\pi}{2}\). Find the maximum and minimum values for P, if any.

Answers (1)

2020-10-27
\(P(t)=100+20\cos6t , 0\leq t\leq \frac{\pi}{20}\leq t\leq\frac{π}{2}\) Differentiating P(t) with respect to t :
\(P′(t)=−20\times 6\times \sin 6t=−120\sin 6t\)
\(P′(t)=0 \geq \sin6t=0\)
\(\sin\oslash =0 \geq \oslash=n\times \pi\)
\(\sin6t=0\geq t=n\frac{\pi}{6}\)
Given 0\leq t\leq \pi /20\leq t\frac{\pi}{2}
So t=0,\frac{\pi}{2},\frac{\pi}{6}t=0,\frac{\pi}{2},\frac{\pi}{6}
On putting the value of t in P(t)we get: When t=0
\(P(0)=100+20\cos (6\times 0)\)
\(P(0)=100+20\times 1\)
\(P(0)=120\)
When t=\frac{\pi}{2} t=\frac{\pi}{2} \(P(\frac{\pi}{2})=100+20\cos(6\times \frac{\pi}{2})\)
\(P(\frac{\pi}{2})=100+20\cos 3\pi\)
\(P\frac{\pi}{2}=100−20\)
\(P\frac{\pi}{2}=80\)
When t=\frac{\pi}{6}t=\frac{\pi}{6}
\(P\frac{\pi}{6}=100+20\cos 6\times \frac{\pi}{6}\)
\(P\frac{\pi}{6}=100−20P\frac{\pi}{6}=100−20\)
\(P\frac{\pi}{6}=80P\frac{\pi}{6}=80\)
Thus we get maximum value of P(t) at t=0 as 120
Thus we get minimum value of P(t) at t=\frac{\pi}{2} and \frac{\pi}{2} and \frac{\pi}{6} as 80
0

Relevant Questions

asked 2021-02-03
Let \(\displaystyle{P}{\left({t}\right)}={100}+{20}{\cos{
asked 2021-02-13
What values(s) of the constant b make \(f(x)=x^{3}−bx\) for \(0\leq x\leq 2\) have:
a.) an absolute min at x=1? Explain.
b.) an absolute max at x=2? Explain.
asked 2021-01-13
find the values of b such that the function has the given maximum or minimum value. f(x) = -x^2+bx-75, Maximum value: 25
asked 2021-01-15
Let \(y(t)=\int_0^tf(t)dt\) If the Laplace transform of y(t) is given \(Y(s)=\frac{19}{(s^2+25)}\) , find f(t)
a) \(f(t)=19 \sin(5t)\)
b) none
c) \(f(t)=6 \sin(2t)\)
d) \(f(t)=20 \cos(6t)\)
e) \(f(t)=19 \cos(5t)\)
asked 2020-10-28
Find the absolute maximum and absolute minimum values of f on the given interval.
\(\displaystyle{f{{\left({t}\right)}}}={5}{t}+{5}{\cot{{\left(\frac{{t}}{{2}}\right)}}},{\left[\frac{\pi}{{4}},{7}\frac{\pi}{{4}}\right]}\)
absolute minimum value-?
absolute maximum value-?
asked 2021-01-22
Question 1
The function \(N(t)=15t\sqrt{t^{2}+1}+2000\) denotes the number of members of a club t months after the club's opening. You can use the following
\(N'(t)=\frac{30t^{2}+15}{\sqrt {t^{2}+1}}\) / and/or \(\int N(t)dt = 2000t+5(t^{2}+1)^{\frac{3}{2})}+ C\)
1.1 How many members were in the club after 12 months?
asked 2020-11-11
Question 1
The function \(\displaystyle{N}{\left({t}\right)}={15}{t}\sqrt{{{t}^{{{2}}}+{1}}}+{2000}\) denotes the number of members of a club t months after the club's opening. You can use the following
\(\displaystyle{N}'{\left({t}\right)}={\frac{{{30}{t}^{{{2}}}+{15}}}{{\sqrt{{{t}^{{{2}}}+{1}}}}}}\) / and/or \(\displaystyle\int{N}{\left({t}\right)}{\left.{d}{t}\right.}={2000}{t}+{5}{\left({t}^{{{2}}}+{1}\right)}^{{{\frac{{{3}}}{{{2}}}}}}\rbrace+{C}\)
1.1 How many members were in the club after 12 months?
asked 2021-02-25
Find f'(a).
\(f(t)= \frac{3t+3}{t+2}\)
asked 2020-12-14
Let C be a circle, and let P be a point not on the circle. Prove that the maximum and minimum distances from P to a point X on C occur when the line X P goes through the center of C. [Hint: Choose coordinate systems so that C is defined by
\(x2 + y2 = r2\) and P is a point (a,0)
on the x-axis with a \(\neq \pm r,\) use calculus to find the maximum and minimum for the square of the distance. Don’t forget to pay attention to endpoints and places where a derivative might not exist.]
asked 2020-12-31
If a cylindrical tank holds 100,000 gallons of water, whichcan be drained from the bottom of the tank in an hour, thenTorricelli's Law gives the volume V of water remaining inthe tank after t minutes as:
V(t)= 100,000( 1- {t/60})2
for: 0 is less than or equal to t which is lessthan or equal to 60
Find the rate at which the water is flowing out of the tank(the instantaneous rate of change of V with respect tot) as a function of t. What are its units? Fortimes t=0, 10, 20, 30, 40, 50, and 60 min, find the flow rate andthea mount of water remaining in the tank. Summarize your findingsin a sentence or two. At what time is the flow rate the greatest?The least?
...